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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def precondition_u_kn(u_kn, N_k, f_k):
 """Subtract a sample-dependent constant from u_kn to improve precision Parameters u_kn : np.ndarray, shape=(n_states, n_samples), dtype='float' The reduced potential energies, i.e. -log unnormalized probabilities N_k : np.ndarray, shape=(n_states), dtype='int' The number of samples in each state f_k : np.ndarray, shape=(n_states), dtype='float' The reduced free energies of each state Returns ------- u_kn : np.ndarray, shape=(n_states, n_samples), dtype='float' The reduced potential energies, i.e. -log unnormalized probabilities Notes ----- Returns u_kn - x_n, where x_n is based on the current estimate of f_k. Upon subtraction of x_n, the MBAR objective function changes by an additive constant, but its derivatives remain unchanged. We choose x_n such that the current objective function value is zero, which should give maximum precision in the objective function. """

    u_kn, N_k, f_k = validate_inputs(u_kn, N_k, f_k)
    u_kn = u_kn - u_kn.min(0)
    u_kn += (logsumexp(f_k - u_kn.T, b=N_k, axis=1)) - N_k.dot(f_k) / float(N_k.sum())
    return u_kn