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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 auth(name, nodes, pcsuser='hacluster', pcspasswd='hacluster', extra_args=None):
    '''
    Ensure all nodes are authorized to the cluster

    name
        Irrelevant, not used (recommended: pcs_auth__auth)
    nodes
        a list of nodes which should be authorized to the cluster
    pcsuser
        user for communication with pcs (default: hacluster)
    pcspasswd
        password for pcsuser (default: hacluster)
    extra_args
        list of extra args for the \'pcs cluster auth\' command

    Example:

    .. code-block:: yaml

        pcs_auth__auth:
            pcs.auth:
                - nodes:
                    - node1.example.com
                    - node2.example.com
                - pcsuser: hacluster
                - pcspasswd: hoonetorg
                - extra_args: []
    '''

    ret = {'name': name, 'result': True, 'comment': '', 'changes': {}}
    auth_required = False

    authorized = __salt__['pcs.is_auth'](nodes=nodes)
    log.trace('Output of pcs.is_auth: %s', authorized)

    authorized_dict = {}
    for line in authorized['stdout'].splitlines():
        node = line.split(':')[0].strip()
        auth_state = line.split(':')[1].strip()
        if node in nodes:
            authorized_dict.update({node: auth_state})
    log.trace('authorized_dict: %s', authorized_dict)

    for node in nodes:
        if node in authorized_dict and authorized_dict[node] == 'Already authorized':
            ret['comment'] += 'Node {0} is already authorized\n'.format(node)
        else:
            auth_required = True
            if __opts__['test']:
                ret['comment'] += 'Node is set to authorize: {0}\n'.format(node)

    if not auth_required:
        return ret

    if __opts__['test']:
        ret['result'] = None
        return ret
    if not isinstance(extra_args, (list, tuple)):
        extra_args = []
    if '--force' not in extra_args:
        extra_args += ['--force']

    authorize = __salt__['pcs.auth'](nodes=nodes, pcsuser=pcsuser, pcspasswd=pcspasswd, extra_args=extra_args)
    log.trace('Output of pcs.auth: %s', authorize)

    authorize_dict = {}
    for line in authorize['stdout'].splitlines():
        node = line.split(':')[0].strip()
        auth_state = line.split(':')[1].strip()
        if node in nodes:
            authorize_dict.update({node: auth_state})
    log.trace('authorize_dict: %s', authorize_dict)

    for node in nodes:
        if node in authorize_dict and authorize_dict[node] == 'Authorized':
            ret['comment'] += 'Authorized {0}\n'.format(node)
            ret['changes'].update({node: {'old': '', 'new': 'Authorized'}})
        else:
            ret['result'] = False
            if node in authorized_dict:
                ret['comment'] += 'Authorization check for node {0} returned: {1}\n'.format(node, authorized_dict[node])
            if node in authorize_dict:
                ret['comment'] += 'Failed to authorize {0} with error {1}\n'.format(node, authorize_dict[node])

    return ret