
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
Let $F_1$, $F_2$ be the left and right foci of the ellipse $\Gamma$: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$), and let $M$ be a point on the ellipse $\Gamma$. If $\triangle MF_1F_2$ is a right triangle and $|MF_1|=2|MF_2|$, then the eccentricity of the ellipse $\Gamma$ is $\_\_\_\_\_\_$.

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