In the triangle $ABC$, we have $\angle ABC = 100^\circ$, $\angle ACB = 65^\circ$, $M \in AB$, $N \in AC$, and $\angle MCB = 55^\circ$, $\angle NBC = 80^\circ$. Find $\angle NMC$.

Let's think step by step and output the final answer within \boxed{}.
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