
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
Twelve crates, each of dimensions $2\mathrm{ft}\times 5\mathrm{ft}\times 7\mathrm{ft}$, are stacked with their orientation chosen at random. If the first crate is placed flat on the floor, and each remaining crate is placed flat on top of the previous one, find the probability that the total height of the stacked crates is exactly $50\mathrm{ft}$. Assume each orientation for a crate is equally likely. Let $\frac{m}{n}$ be this probability, with $m$ and $n$ relatively prime. Find $m$.

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