
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 hyperbola \( C_{1}: 2x^{2} - y^{2} = 1 \) and the ellipse \( C_{2}: 4x^{2} + y^{2} = 1 \), let \( M \) and \( N \) be moving points on the hyperbola \( C_{1} \) and the ellipse \( C_{2} \) respectively, with \( O \) as the origin. If \( O M \) is perpendicular to \( O N \), find the distance from point \( O \) to the line \( M N \).

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