
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 circle $C$: $(x - \sqrt{3})^2 + (y - 1)^2 = 1$ and two points $A(-t, 0)$, $B(t, 0)$, $(t > 0)$, find the maximum value of $t$ such that there exists a point $P$ on the circle such that $\angle APB = 90^{\circ}$.

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