
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
The perimeter of triangle \(ABC\) is \(2p\). Points \(M\) and \(N\) are taken on sides \(AB\) and \(AC\) such that \(MN \parallel BC\) and \(MN\) is tangent to the inscribed circle of triangle \(ABC\). Find the maximum length of segment \(MN\).

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