
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
Ten identical crates each with dimensions $2\mathrm{ft}\times 3\mathrm{ft}\times 5\mathrm{ft}$ are to be stacked one on top of the other. Each crate's orientation is chosen at random, and placed flat on the previous crate. Calculate the probability that the total height of the stack is exactly $38\mathrm{ft}$. Express the probability in its simplest form, $\frac{m}{n}$, where $m$ and $n$ are coprime integers. Find $m$.

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