
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 right triangle \( \triangle ABC \), \(\angle C = 90^\circ\), and \( AC = 11 \). Point \( D \) is on side \( AC \) such that \( AD = 10 \) and \(\angle BDC = 3 \angle BAC\). If the perimeter of \(\triangle ABC\) can be written in the form \(\frac{1}{2}(m + \sqrt{n})\) (where \( m \) and \( n \) are integers), find the value of \( m + n \).

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