
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, two cuts are made starting at a distance of $\sqrt{20}$ 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. When two edges of a cut meet, they are taped together, forming a paper tray with inclined sides. Determine the height of the tray, which is the perpendicular distance from the base of the tray to the plane formed by the upper edges.

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