
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]


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Consider the hyperbola $M$: $x^2-\frac{y^2}{b^2}=1$ with left and right foci $F_1$ and $F_2$, respectively. Denote $|F_1F_2|=2c$. Let $O$ be the origin of the coordinate system, and let $c$ be the radius of a circle centered at $O$. The intersection point $P$ of the circle and the hyperbola $M$ in the first quadrant satisfies $|PF_1|=c+2$. Find the $x$-coordinate of point $P$.

A) $\frac{\sqrt{3}+1}{2}$
B) $\frac{\sqrt{3}+2}{2}$
C) $\frac{\sqrt{3}+3}{2}$
D) $\frac{3\sqrt{3}}{2}$

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