
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 $\frac{{x}^{2}}{16}+\frac{{{y}^{2}}}{{{m}^{2}}}=1(m > 0)$, if the projection of one of the intersection points $M$ of the line $y=\frac{\sqrt{2}}{2}x$ and the ellipse on the $x$-axis is exactly the right focus $F$ of the ellipse, what is the eccentricity of the ellipse?

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