$1.(10$ points$)$ Two firms compete in one market. Their cost functions are C$1($y$1)$ and C$2($y$2)$ respectively, with increasing marginal cost. Products are homogeneous, so the total market supply is Y $=$ y$1+$ y$2.$ The downward sloping market $($inverse$)$ demand function is P $=$ P $($Y $).$

$($a$)$ If these two firms compete following Cournot competition, write down their optimization problem and the associated first order conditions $($no need to solve it$).$