
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 lateral sides \(AB\) and \(CD\) of the trapezoid \(ABCD\), points \(M\) and \(N\) are taken such that segment \(MN\) is parallel to the bases and divides the area of the trapezoid in half. Find the length of \(MN\) if \(BC = a\) and \(AD = b\).

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