
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
There are twelve identical crates each with dimensions $3\mathrm{ft}\times 4\mathrm{ft}\times 5\mathrm{ft}$. The first crate is placed flat on the floor, and each of the remaining eleven crates is placed, in turn, flat on top of the previous crate, with the orientation of each crate chosen at random. Determine the probability that the stack of crates is exactly $50\mathrm{ft}$ tall. Let $\frac {m}{n}$ be this probability, where $m$ and $n$ are relatively prime positive integers. Find $m$.

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