
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
Three lines intersect at a point \( O \). A point \( M \) is taken outside these lines, and perpendiculars are dropped from it to the lines. Points \(\mathrm{H}_{1}, \mathrm{H}_{2}, \mathrm{H}_{3}\) are the feet of these perpendiculars. Find the ratio of the length \( OM \) to the radius of the circumscribed circle around triangle \(\mathrm{H}_{1} \mathrm{H}_{2} \mathrm{H}_{3}\).

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