
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 an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. Let $F_{1}$ be the center of a circle with radius $|F_{1}F_{2}|$, intersecting the $y$-axis at points $A$ and $B$ (both points $A$ and $B$ are outside of $C$). Connect $F_{1}A$ with $C$ at point $P$. If $\overrightarrow{{F}_{2}P}\cdot \overrightarrow{{F}_{2}B}=0$, then $\angle AF_{1}B=$____; the eccentricity of the ellipse $C$ is ____.

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