I have this Cournot game in which $n$ firms produce quantities $q_1, \ldots, q_n$ with respective marginal costs $c_1, \ldots, c_n$. They all sell at price $P=1-(q_1 + \cdots + q_n)$. For any $i$ the marginal cost $c_i$ has equal chance being $0$ or $c$ where $0<c<\frac{4}{n+3}$, and each firm $i$ knows $c_i$ only. I need to find a symmetric best response.
My attempt so far is to express the profit of firm $i$ as $(P-c_i)q_i$, take the derivative, set to $0$. This results in $1-(q_1+\cdots+2q_i+\cdots+q_n)-c_i=0$. I could then make a matrix equation and possibly try to solve it, but this seems like I have too many unknown quantities. Something doesn't quite feel right.
Since the solution is supposed to be symmetric I'm wondering if I instead should have reasoned that for all $i,j$ we have $q_i=q_j$ and therefore profit is $(1-nq_i - c_i)q_i$ and again maximize that. If I do, I get $q_i=\frac{1-c_i}{2n}$. I'm not sure if that's the answer or if I should do more ... at no point do I seem to have used the probabilities with which firms believe other firms to have their marginal cost.
Any insight would be appreciated.