
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
On the sides of angle \(AOB\) from vertex \(O\), segments \(OA\) and \(OB\) are laid out such that \(OA > OB\). A point \(M\) is taken on segment \(OA\), and point \(N\) is taken on the extension of segment \(OB\) such that \(AM = BN = x\). Find the value of \(x\) for which segment \(MN\) has the smallest length.

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