
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 $150$. From each corner, a wedge is cut in the following manner: at each corner, the starting points of the two cuts are each at a distance $\sqrt{50}$ from the corner, and they meet on the diagonal at an angle of $45^\circ$. When folded and the edges of the cuts meet, they are taped together, forming a tray whose perpendicular height from the base to the uplifted edges needs to be calculated and expressed in the simplest radical form.

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