
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
Points \(M\) and \(N\) are located on the lateral sides \(AB\) and \(CD\) of trapezoid \(ABCD\) respectively, with \(MN \parallel AD\). It is known that the area of trapezoid \(MBCN\) is in the ratio \(2:3\) to the area of trapezoid \(AMND\). Find the length \(MN\) if \(BC = a\) and \(AD = b\).

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