
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 triangle \( ABC \), the angle bisectors \( BB_1 \) (internal) and \( BB_2 \) (external) are drawn at vertex \( B \). From the orthocenter \( H \), perpendiculars \( HH_1 \) and \( HH_2 \) are dropped to the lines \( BB_1 \) and \( BB_2 \), respectively. In what ratio does the line \( H_1H_2 \) divide the side \( AC \)?

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