
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
Knowing that segment \( CD \) has a length of 6 and its midpoint is \( M \), two triangles \( \triangle ACD \) and \( \triangle BCD \) are constructed on the same side with \( CD \) as a common side and both having a perimeter of 16, such that \( \angle AMB = 90^{\circ} \). Find the minimum area of \( \triangle AMB \).

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