
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 equation of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$, with the major axis $A_{1}A_{2}$, a perpendicular line is drawn from a point $M$ on the ellipse to the $x$-axis, with the foot of the perpendicular being $P$, determine the eccentricity of the ellipse given that $\frac{|MP|^2}{|A_1P| \cdot |A_2P|}=\frac{1}{3}$.

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