
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 that the center of the ellipse is at the origin, the foci lie on the coordinate axis, and the line \( y = x + 1 \) intersects the ellipse at points \( P \) and \( Q \). Additionally, \( OP \perp OQ \) and \( |PQ| = \frac{\sqrt{10}}{2} \). Find the equation of the ellipse.

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