
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, make two cuts starting at a distance of $8$ from the corner, and have them meet on the diagonal at an angle of $45^{\circ}$. After making the cuts, fold the paper such that the edges along the cuts meet and form a tray with slanted sides. Determine the exact height of this tray, given that the height can be expressed as $\sqrt[n]{m}$, where $m<1000$, and is not divisible by the $n$th power of any prime. Find $m + n$.

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