Is it possible that the system \begin{equation} \begin{cases} 2\dot{q}(t) + \dot{q}(t1) + \dot{q}(t+1) = c & \text{if} \hspace{5mm} 0 \le t \le 2 , \dot{q}(t) + \dot{q}(t1) = k & \text{if} \hspace{5mm} 2 \le t \le 3 \end{cases} \end{equation} has, for suitable constants $c$ and $k$, any $\mathcal{C}^2$ solutions $q:[1,3] \to \mathbb{R}$ satisfying the conditions $q(t) =  t$ for $t \in [1, 0]$ and $q(3)=2$ ?
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