
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 two points $A(a,0)$, $B(-a,0)$ $(a > 0)$, if there exists a point $P$ on the curve $x^{2}+y^{2}-4\sqrt{3}x-4y+7=0$ such that $\angle APB=90^{\circ}$, determine the range of the positive real number $a$.

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