
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 an ellipse centered at the origin with its foci on the x-axis and an eccentricity $e = \frac{\sqrt{2}}{2}$. A chord that passes through the right focus of the ellipse and is perpendicular to the major axis has a length of $\sqrt{2}$.

(1) Find the standard equation of the ellipse;  
(2) Given a line $l$ intersects the ellipse at points $P$ and $Q$, with $O$ being the origin and $OP \perp OQ$, investigate whether the distance from point $O$ to line $l$ is constant. If it is, find this constant distance; if not, explain why.

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