
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 such that each cut starts at a distance $\sqrt{20}$ from the corner and meets the diagonal of the square at an angle of $45^\circ$. The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The resulting paper tray has sides that are not at right angles to the base. Determine the height of the tray, expressed in the form $\sqrt[n]{m}$, where $m$ and $n$ are positive integers, $m<1000$, and $m$ is not divisible by the $n$th power of any prime. Find $m+n$.

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