
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
The center of an ellipse is at the origin, and its foci are located on the coordinate axis. The line $y=x+1$ intersects the ellipse at points $P$ and $Q$. If $\overset{→}{OP} \cdot \overset{→}{OQ}=0$ and $| \overset{→}{PQ}|= \frac{ \sqrt{10}}{2}$, find the equation of the ellipse.

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