
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 piece of paper has sides of length $120$. From each corner, a wedge is cut where each of the two cuts for the wedge starts at a distance $\sqrt{20}$ from the corner, meeting on the diagonal at an angle of $45^\circ$. After the cuts, the paper is folded up along the lines joining the vertices of adjacent cuts, and the edges are taped together to form a tray with slanted sides. Compute the height of the tray, given vertically from the base to the top, expressed in simplified radical form. Find $m$ such that the height of the tray can be expressed as $\sqrt{m}$, and where $m$ is the smallest integer not divisible by any square number greater than 1.

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