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/* mpc_rootofunity -- primitive root of unity.
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Copyright (C) 2012, 2016 INRIA
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This file is part of GNU MPC.
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GNU MPC is free software; you can redistribute it and/or modify it under
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the terms of the GNU Lesser General Public License as published by the
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Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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GNU MPC is distributed in the hope that it will be useful, but WITHOUT ANY
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WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for
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more details.
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You should have received a copy of the GNU Lesser General Public License
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along with this program. If not, see http://www.gnu.org/licenses/ .
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*/
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#include <stdio.h> /* for MPC_ASSERT */
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#include "mpc-impl.h"
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static unsigned long
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gcd (unsigned long a, unsigned long b)
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{
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if (b == 0)
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return a;
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else return gcd (b, a % b);
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}
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/* put in rop the value of exp(2*i*pi*k/n) rounded according to rnd */
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int
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mpc_rootofunity (mpc_ptr rop, unsigned long n, unsigned long k, mpc_rnd_t rnd)
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{
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unsigned long g;
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mpq_t kn;
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mpfr_t t, s, c;
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mpfr_prec_t prec;
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int inex_re, inex_im;
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mpfr_rnd_t rnd_re, rnd_im;
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if (n == 0) {
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/* Compute exp (0 + i*inf). */
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mpfr_set_nan (mpc_realref (rop));
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mpfr_set_nan (mpc_imagref (rop));
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return MPC_INEX (0, 0);
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}
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/* Eliminate common denominator. */
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k %= n;
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g = gcd (k, n);
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k /= g;
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n /= g;
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/* Now 0 <= k < n and gcd(k,n)=1. */
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/* We assume that only n=1, 2, 3, 4, 6 and 12 may yield exact results
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and treat them separately; n=8 is also treated here for efficiency
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reasons. */
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if (n == 1)
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{
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/* necessarily k=0 thus we want exp(0)=1 */
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MPC_ASSERT (k == 0);
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return mpc_set_ui_ui (rop, 1, 0, rnd);
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}
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else if (n == 2)
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{
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/* since gcd(k,n)=1, necessarily k=1, thus we want exp(i*pi)=-1 */
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MPC_ASSERT (k == 1);
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return mpc_set_si_si (rop, -1, 0, rnd);
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}
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else if (n == 4)
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{
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/* since gcd(k,n)=1, necessarily k=1 or k=3, thus we want
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exp(2*i*pi/4)=i or exp(2*i*pi*3/4)=-i */
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MPC_ASSERT (k == 1 || k == 3);
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if (k == 1)
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return mpc_set_ui_ui (rop, 0, 1, rnd);
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else
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return mpc_set_si_si (rop, 0, -1, rnd);
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}
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else if (n == 3 || n == 6)
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{
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MPC_ASSERT ((n == 3 && (k == 1 || k == 2)) ||
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(n == 6 && (k == 1 || k == 5)));
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/* for n=3, necessarily k=1 or k=2: -1/2+/-1/2*sqrt(3)*i;
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for n=6, necessarily k=1 or k=5: 1/2+/-1/2*sqrt(3)*i */
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inex_re = mpfr_set_si (mpc_realref (rop), (n == 3 ? -1 : 1),
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MPC_RND_RE (rnd));
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/* inverse the rounding mode for -sqrt(3)/2 for zeta_3^2 and zeta_6^5 */
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rnd_im = MPC_RND_IM (rnd);
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if (k != 1)
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rnd_im = INV_RND (rnd_im);
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inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 3, rnd_im);
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mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
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if (k != 1)
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{
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mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
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inex_im = -inex_im;
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}
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return MPC_INEX (inex_re, inex_im);
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}
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else if (n == 12)
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{
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/* necessarily k=1, 5, 7, 11:
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k=1: 1/2*sqrt(3) + 1/2*I
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k=5: -1/2*sqrt(3) + 1/2*I
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k=7: -1/2*sqrt(3) - 1/2*I
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k=11: 1/2*sqrt(3) - 1/2*I */
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MPC_ASSERT (k == 1 || k == 5 || k == 7 || k == 11);
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/* inverse the rounding mode for -sqrt(3)/2 for zeta_12^5 and zeta_12^7 */
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rnd_re = MPC_RND_RE (rnd);
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if (k == 5 || k == 7)
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rnd_re = INV_RND (rnd_re);
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inex_re = mpfr_sqrt_ui (mpc_realref (rop), 3, rnd_re);
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inex_im = mpfr_set_si (mpc_imagref (rop), k < 6 ? 1 : -1,
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MPC_RND_IM (rnd));
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mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
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if (k == 5 || k == 7)
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{
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mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
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inex_re = -inex_re;
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}
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return MPC_INEX (inex_re, inex_im);
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}
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else if (n == 8)
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{
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/* k=1, 3, 5 or 7:
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k=1: (1/2*I + 1/2)*sqrt(2)
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k=3: (1/2*I - 1/2)*sqrt(2)
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k=5: -(1/2*I + 1/2)*sqrt(2)
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k=7: -(1/2*I - 1/2)*sqrt(2) */
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MPC_ASSERT (k == 1 || k == 3 || k == 5 || k == 7);
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rnd_re = MPC_RND_RE (rnd);
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if (k == 3 || k == 5)
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rnd_re = INV_RND (rnd_re);
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rnd_im = MPC_RND_IM (rnd);
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if (k > 4)
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rnd_im = INV_RND (rnd_im);
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inex_re = mpfr_sqrt_ui (mpc_realref (rop), 2, rnd_re);
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inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 2, rnd_im);
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mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
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if (k == 3 || k == 5)
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{
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mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
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inex_re = -inex_re;
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}
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if (k > 4)
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{
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mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
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inex_im = -inex_im;
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}
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return MPC_INEX (inex_re, inex_im);
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}
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prec = MPC_MAX_PREC(rop);
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/* For the error analysis justifying the following algorithm,
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see algorithms.tex. */
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mpfr_init2 (t, 67);
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mpfr_init2 (s, 67);
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mpfr_init2 (c, 67);
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mpq_init (kn);
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mpq_set_ui (kn, k, n);
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mpq_mul_2exp (kn, kn, 1); /* kn=2*k/n < 2 */
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do {
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prec += mpc_ceil_log2 (prec) + 5; /* prec >= 6 */
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mpfr_set_prec (t, prec);
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mpfr_set_prec (s, prec);
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mpfr_set_prec (c, prec);
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mpfr_const_pi (t, MPFR_RNDN);
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mpfr_mul_q (t, t, kn, MPFR_RNDN);
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mpfr_sin_cos (s, c, t, MPFR_RNDN);
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}
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while ( !mpfr_can_round (c, prec - (4 - mpfr_get_exp (c)),
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MPFR_RNDN, MPFR_RNDZ,
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MPC_PREC_RE(rop) + (MPC_RND_RE(rnd) == MPFR_RNDN))
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|| !mpfr_can_round (s, prec - (4 - mpfr_get_exp (s)),
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MPFR_RNDN, MPFR_RNDZ,
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MPC_PREC_IM(rop) + (MPC_RND_IM(rnd) == MPFR_RNDN)));
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inex_re = mpfr_set (mpc_realref(rop), c, MPC_RND_RE(rnd));
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inex_im = mpfr_set (mpc_imagref(rop), s, MPC_RND_IM(rnd));
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mpfr_clear (t);
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mpfr_clear (s);
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mpfr_clear (c);
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mpq_clear (kn);
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return MPC_INEX(inex_re, inex_im);
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}
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