
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
Suppose that the angles of triangle $DEF$ satisfy
\[
\cos 3D + \cos 3E + \cos 3F = 1.
\]
One side of the triangle has length 12, and the triangle's perimeter is 40. Find the maximum possible length for another side of the triangle.

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