
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 line \( l \) with a slope of \(\frac{\sqrt{2}}{2}\) intersects the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)\) at two distinct points \( P \) and \( Q \). If the projections of points \( P \) and \( Q \) on the \(x\)-axis are exactly the foci of the ellipse, find the eccentricity of the ellipse.

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