
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
Let \( \triangle ABC \) be a triangle with \( AB = 3 \), \( AC = 8 \), \( BC = 7 \). \( M \) and \( N \) are the midpoints of \( \overline{AB} \) and \( \overline{AC} \), respectively. Point \( T \) is selected on side \( BC \) such that \( AT = TC \). The circumcircles of triangles \( BAT \) and \( MAN \) intersect at \( D \). Compute \( DC \).

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