
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 hyperbola $C_{1}: \frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1 \ (a>0,\ b>0)$ intersecting the circle $C_{2}: x^{2}+y^{2}=c^{2}$ (where $c$ is half the focal distance of the hyperbola) at point P in the first quadrant. If $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C_{1}$, respectively, and $\angle PF_{2}F_{1}= \frac {\pi}{3}$, then the eccentricity of the hyperbola is $\boxed{\sqrt {3}+1}$.

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