
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 such that each of the two cuts for the wedge starts at a distance of $2\sqrt{23}$ from the corner, and they meet on the diagonal at an angle of $45^{\circ}$. After cutting, the paper is folded along the lines joining the ends of adjacent cuts into the corners. When the edges of a cut meet, they are taped together to form a truncated pyramid shape. Determine the height of the pyramid, that is, the perpendicular distance from the base of the paper to the apex of the pyramid, expressed as $\sqrt[n]{m}$ where $m$ and $n$ are positive integers, with $m < 2000$, 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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