
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
A square plate has side length  $L$  and negligible thickness. It is laid down horizontally on a table and is then rotating about the axis  $\overline{MN}$  where  $M$  and  $N$  are the midpoints of two adjacent sides of the square. The moment of inertia of the plate about this axis is  $kmL^2$ , where  $m$  is the mass of the plate and  $k$  is a real constant. Find  $k$ . 

<span style="color:red">Diagram will be added to this post very soon. If you want to look at it temporarily, see the PDF.</span>

*Problem proposed by Ahaan Rungta*

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