The reaction rate of the triple-alpha reaction can be enhanced in hot and dense environments due to the deexcitation of the Hoyle state in ¹²C by neutrons. The cross section of the deexcitation should be determined for the enhanced reaction rate. We plan to obtain the cross section by measuring the inverse reaction using a neutron beam around 10 MeV and an active target system. In the present paper, we report a proof-of-principle experiment using a neutron beam at 14 MeV. The obtained cross section is consistent with a previous result, demonstrating the validity of our method.