$2.$ Peter finds himself trapped in Multiverse, which has three open portals. Behind one of the portals is a way back home, and behind the other two is Dark Dimension. Walking into Dark Dimension means certain death by Dormammu, the Destroyer of Worlds. On each portal there is an inscription:

$-$ On portal $1$ it says: "Home is behind this portal."

$-$ On portal $2$ it says: "Home is not behind this portal."

$-$ On portal $3$ it says: $"$If Home is not behind portal $1$ then it is behind this portal."

Peter also finds a note with the following information: at least one of the three inscriptions is true, and at least one of them is false.

$($a$)$ Let $\backslash ($ P$\_\{$i$\}\backslash )$ be the proposition "Home is behind portal $\backslash ($ i $\backslash )",$ for $\backslash ($ i$=1,2,3\backslash ).$ Translate the note Peter found into a propositional statement built out of $\backslash ($ P$\_\{1\},$ P$\_\{2\}\backslash )$ and $\backslash ($ P$\_\{3\}\backslash ).$

$($b$)$ Using truth tables, determine which portal will lead Peter to home.