
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given point $P(-1,1)$, let $C$ be the family of circles described by the equation $(x-t)^{2}+(y-t)^{2}=1$, where $t \in R$. Let $A$ and $B$ be the points of tangency of the tangent lines to circle $C$ that pass through point $P$. Find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$.

Present the answer in LaTex format: \boxed{Your answer}
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