
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 point $P(n,4)$ is on the ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a > b > 0)$, and $F_1$, $F_2$ are the two foci of the ellipse $C$, and the diameter of the incircle of $\triangle PF_1F_2$ is $3$, determine the eccentricity of this ellipse.

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