
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 of the two cuts starts at a distance of $5$ from the corner and they meet on the diagonal at an angle of $45^{\circ}$. The paper is then folded up along the lines joining the vertices of adjacent cuts to form a tray with inclined sides. Determine the height of the tray, defined as the perpendicular distance from the base to the upper edge formed after folding.

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