
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 hyperbola $C: \frac{x^{2}}{4} - \frac{y^{2}}{3} = 1$, with its right vertex at $P$.
(1) Find the standard equation of the circle centered at $P$ and tangent to both asymptotes of the hyperbola $C$;
(2) Let line $l$ pass through point $P$ with normal vector $\overrightarrow{n}=(1,-1)$. If there are exactly three points $P_{1}$, $P_{2}$, and $P_{3}$ on hyperbola $C$ with the same distance $d$ to line $l$, find the value of $d$.

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