
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
In an acute-angled triangle \( ABC \), the altitude \( AA_1 \) is drawn. \( H \) is the orthocenter of triangle \( ABC \). It is known that \( AH = 3 \), \( A_1H = 2 \), and the radius of the circumcircle of triangle \( ABC \) is 4. Find the distance from the center of this circumcircle to \( H \).

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