
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
Convex quadrilateral \( MATH \) is given with \( \frac{HM}{MT} = \frac{3}{4} \), and \( \angle ATM = \angle MAT = \angle AHM = 60^\circ \). \( N \) is the midpoint of \( MA \), and \( O \) is a point on \( TH \) such that lines \( MT, AH, NO \) are concurrent. Find the ratio \( \frac{HO}{OT} \).

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