
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
The left and right foci of the hyperbola $x^{2}-\frac{y^{2}}{b^{2}}=1$ are denoted as $F_{1}$ and $F_{2}$, respectively, with $|F_{1}F_{2}|=2c$. A circle with the origin $O$ as its center and $c$ as its radius intersects the hyperbola $M$ at point $P$ in the first quadrant. If $|PF_{1}|=c+2$, calculate the x-coordinate of point $P$.

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