Find the number of sequences $a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9,a_{10},a_{11},a_{12}$ which fulfill the following conditions: $$ a_k^2=a_k,\quad \sum\limits_{i=1}^{12} a_i=7,\quad \sum\limits_{i=1}^k a_i\geq \frac{k}{2}\quad $$ for every $1\leq k\leq 12$.
I'm struggling to find a catalan-numbers-approach to this problem. (if there exists)