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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 solve_mbar_once(u_kn_nonzero, N_k_nonzero, f_k_nonzero, method="hybr", tol=1E-12, options=None):
 """Solve MBAR self-consistent equations using some form of equation solver. Parameters u_kn_nonzero : np.ndarray, shape=(n_states, n_samples), dtype='float' The reduced potential energies, i.e. -log unnormalized probabilities for the nonempty states N_k_nonzero : np.ndarray, shape=(n_states), dtype='int' The number of samples in each state for the nonempty states f_k_nonzero : np.ndarray, shape=(n_states), dtype='float' The reduced free energies for the nonempty states method : str, optional, default="hybr" The optimization routine to use. This can be any of the methods available via scipy.optimize.minimize() or scipy.optimize.root(). tol : float, optional, default=1E-14 The convergance tolerance for minimize() or root() verbose: bool Whether to print information about the solution method. options: dict, optional, default=None Optional dictionary of algorithm-specific parameters. See scipy.optimize.root or scipy.optimize.minimize for details. Returns ------- f_k : np.ndarray The converged reduced free energies. results : dict Dictionary containing entire results of optimization routine, may be useful when debugging convergence. Notes ----- This function requires that N_k_nonzero > 0--that is, you should have already dropped all the states for which you have no samples. Internally, this function works in a reduced coordinate system defined by subtracting off the first component of f_k and fixing that component to be zero. For fast but precise convergence, we recommend calling this function multiple times to polish the result. `solve_mbar()` facilitates this. """

    u_kn_nonzero, N_k_nonzero, f_k_nonzero = validate_inputs(u_kn_nonzero, N_k_nonzero, f_k_nonzero)
    f_k_nonzero = f_k_nonzero - f_k_nonzero[0]  # Work with reduced dimensions with f_k[0] := 0
    u_kn_nonzero = precondition_u_kn(u_kn_nonzero, N_k_nonzero, f_k_nonzero)

    pad = lambda x: np.pad(x, (1, 0), mode='constant')  # Helper function inserts zero before first element
    unpad_second_arg = lambda obj, grad: (obj, grad[1:])  # Helper function drops first element of gradient

    # Create objective functions / nonlinear equations to send to scipy.optimize, fixing f_0 = 0
    grad = lambda x: mbar_gradient(u_kn_nonzero, N_k_nonzero, pad(x))[1:]  # Objective function gradient
    grad_and_obj = lambda x: unpad_second_arg(*mbar_objective_and_gradient(u_kn_nonzero, N_k_nonzero, pad(x)))  # Objective function gradient and objective function
    hess = lambda x: mbar_hessian(u_kn_nonzero, N_k_nonzero, pad(x))[1:][:, 1:]  # Hessian of objective function

    with warnings.catch_warnings(record=True) as w:
        if method in ["L-BFGS-B", "dogleg", "CG", "BFGS", "Newton-CG", "TNC", "trust-ncg", "SLSQP"]:
            if method in ["L-BFGS-B", "CG"]:
                hess = None  # To suppress warning from passing a hessian function.
            results = scipy.optimize.minimize(grad_and_obj, f_k_nonzero[1:], jac=True, hess=hess, method=method, tol=tol, options=options)
            f_k_nonzero = pad(results["x"])
        elif method == 'adaptive':
            results = adaptive(u_kn_nonzero, N_k_nonzero, f_k_nonzero, tol=tol, options=options)
            f_k_nonzero = results # they are the same for adaptive, until we decide to return more.
        else:
            results = scipy.optimize.root(grad, f_k_nonzero[1:], jac=hess, method=method, tol=tol, options=options)
            f_k_nonzero = pad(results["x"])

    # If there were runtime warnings, show the messages
    if len(w) > 0:
        can_ignore = True
        for warn_msg in w:
            if "Unknown solver options" in str(warn_msg.message):
                continue
            warnings.showwarning(warn_msg.message, warn_msg.category,
                                 warn_msg.filename, warn_msg.lineno, warn_msg.file, "")
            can_ignore = False  # If any warning is not just unknown options, can ]not skip check
        if not can_ignore:
            # Ensure MBAR solved correctly
            w_nk_check = mbar_W_nk(u_kn_nonzero, N_k_nonzero, f_k_nonzero)
            check_w_normalized(w_nk_check, N_k_nonzero)
            print("MBAR weights converged within tolerance, despite the SciPy Warnings. Please validate your results.")

    return f_k_nonzero, results