
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
Two circles touch each other externally at point \(K\). One line touches these circles at different points \(A\) and \(B\), and another line touches them at different points \(C\) and \(D\). The common tangent to the circles, passing through point \(K\), intersects these lines at points \(M\) and \(N\). Find \(MN\) if \(AC = a\) and \(BD = b\).

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