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Let
$\mathcal{P}=\left\{(x, y): x+y \geq 1,2 x+y \geq 2, x+2 y \geq 2,(x-1)^{2}+(y-1)^{2} \leq 1\right\} .$
Compute
$\min _{(x, y) \in \mathcal{P}} 2 x+3 y$

1. $2$
2. $3$
3. $4$
4. $6$
5. None of the above