
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$, where the left and right vertices are $A$ and $B$ respectively, the right focus is $F$, $P$ is a point on the ellipse, the line $AP$ intersects the line at point $M$, the angle bisector of $\angle PFB$ intersects the line $x=a$ at point $N$. If $PF\perp AB$, the area of triangle $MAB$ is $6$ times the area of triangle $NFB$, then the eccentricity of the ellipse $C$ is ____.

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