
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 a circle $C: (x-1)^{2} + (y-1)^{2} = 4$ and two points $A(a,0)$, $B(-a,0)$ where $a > 0$, find the value of real number $a$ such that there exists only one point $P$ on circle $C$ such that $\angle APB = 90^{\circ}$.

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