
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 chosen such that the segment \(MN\) is parallel to the bases and divides the area of the trapezoid in half. Find the length of \(MN\), given that \(BC = a\) and \(AD = b\).

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