
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 the acute-angled triangle $ABC$, where $\angle B > \angle C$, $H$, $I$, and $O$ are the orthocenter, incenter, and circumcenter, respectively, and $M$ is the midpoint of $AI$. Line $AD$ is drawn perpendicular to $BC$ at point $D$, and it satisfies the condition $\angle DAM = 2 \angle ADM$. Express $\angle HIO$ in terms of $\angle B$.

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