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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 installed(name,
              cyg_arch='x86_64',
              mirrors=None):
    '''
    Make sure that a package is installed.

    name
        The name of the package to install

    cyg_arch : x86_64
        The cygwin architecture to install the package into.
        Current options are x86 and x86_64

    mirrors : None
        List of mirrors to check.
        None will use a default mirror (kernel.org)

    CLI Example:

    .. code-block:: yaml

        rsync:
          cyg.installed:
            - mirrors:
              - http://mirror/without/public/key: ""
              - http://mirror/with/public/key: http://url/of/public/key
    '''
    ret = {'name': name, 'result': None, 'comment': '', 'changes': {}}

    if cyg_arch not in ['x86', 'x86_64']:
        ret['result'] = False
        ret['comment'] = 'The \'cyg_arch\' argument must\
 be one of \'x86\' or \'x86_64\''
        return ret

    LOG.debug('Installed State: Initial Mirror list: %s', mirrors)

    if not __salt__['cyg.check_valid_package'](name,
                                               cyg_arch=cyg_arch,
                                               mirrors=mirrors):
        ret['result'] = False
        ret['comment'] = 'Invalid package name.'
        return ret

    pkgs = __salt__['cyg.list'](name, cyg_arch)
    if name in pkgs:
        ret['result'] = True
        ret['comment'] = 'Package is already installed.'
        return ret

    if __opts__['test']:
        ret['comment'] = 'The package {0} would\
 have been installed'.format(name)
        return ret

    if __salt__['cyg.install'](name,
                               cyg_arch=cyg_arch,
                               mirrors=mirrors):
        ret['result'] = True
        ret['changes'][name] = 'Installed'
        ret['comment'] = 'Package was successfully installed'
    else:
        ret['result'] = False
        ret['comment'] = 'Could not install package.'

    return ret