
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 $120$ cm. From each corner, a wedge is cut where each cut starts at a distance of $5$ cm from the corner, and the cuts meet on the diagonal at an angle of $45^{\circ}$. After the cuts meet, the paper is folded up along the lines joining the vertices of adjacent cuts to create a tray. Determine the height of the tray, which is defined as the perpendicular distance between the plane of the base and the plane formed by the upper edges. Express your answer in the simplest radical form.

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