
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 ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, points $A$ and $F$ are the left vertex and left focus of the ellipse $C$, respectively. Point $P$ is a moving point on the circle $⊙O$: $x^{2}+y^{2}=b^{2}$. If $\frac{|AP|}{|FP|}$ is a constant, then the eccentricity of the ellipse $C$ is _______.

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