
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 internal bisector $AL$ ($L$ belongs to side $BC$), the altitude $BH$ ($H$ belongs to side $AC$), and the median $CM$ ($M$ belongs to side $AB$) are drawn.

It is known that the angles $\angle CAL$, $\angle ABH$, and $\angle BCM$ are equal.

Determine, with reasoning, the measures of the angles of triangle $ABC$.

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