Chance
Opportunity, Turning Point
noun
けいき
契機 (けいき) Chance, Opportunity, Turning Point
刑期 (けいき) Prison Term
機 (き) Machine, Chance, Opportunity
機会 (きかい) Opportunity, Chance
It's his last Chance 契機 for cake ケーキ before his Prison Term 刑期 begins.
He pledged 契 himself to the machines 機 because they said there was a Chance 契機 for cake ケーキ (just like in the Portal games).