
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
A triangle \(ABC\) has side lengths \(AB = 50 \, \text{cm}\), \(BC = 20 \, \text{cm}\), and \(AC = 40 \, \text{cm}\). Let \(M\) and \(N\) be points on side \(AB\) such that \(CM\) is the angle bisector of \(\angle ACB\) and \(CN\) is the altitude to side \(AB\). What is the length, in centimeters, of \(MN\)?

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