
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 parabola \(C_{1}: x^{2}=2 p y\) has a focus at \(F\). The hyperbola \(C_{2}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) has foci \(F_{1}\) and \(F_{2}\). Point \(P\) is a common point of the two curves in the first quadrant. If the points \(P\), \(F\), and \(F_{1}\) are collinear and there is a common tangent to \(C_{1}\) and \(C_{2}\) at \(P\), find the eccentricity of \(C_{2}\).

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