
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 rectangular piece of paper has dimensions $100$ by $150$. From each corner of the rectangle, a wedge is cut such that the two cuts for each wedge start $7$ units from the corner, and they meet on the diagonal of the wedge at a $45^{\circ}$ angle. The paper is then folded along the lines joining the vertices of adjacent cuts, and when the two edges of a cut meet, they are taped together, forming a rectangular tray. Determine the height of the tray, which is the perpendicular distance from the plane of the base to the plane formed by the upper edges. Given that the height can be expressed as $\sqrt[n]{m}$, where $m$ and $n$ are positive integers, $m < 1000$, 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}
user