
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 \( H \) and \( O \) are the orthocenter and circumcenter of the acute triangle \( \triangle ABC \) respectively, and \( OD \perp BC \) with \( D \) being the foot of the perpendicular from \( O \) to \( BC \), find the ratio \( \frac{AH}{OD} \).

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