
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 the circle \(C\): \((x-t)^{2}+(y-t+2)^{2}=1\) with \(t \in \mathbb{R}\), construct tangents to the circle passing through the point \(P(-1,1)\). If the points of tangency are denoted by \(A\) and \(B\), find the minimum value of \( \vec{PA} \cdot \vec{PB}\).

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