
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 identical crates each with dimensions $2\mathrm{ft}\times 5\mathrm{ft}\times 7\mathrm{ft}$. The first crate is placed flat on the floor. Each of the remaining eleven crates is placed, in turn, atop the previous crate with orientation selected at random. Let $\frac {m}{n}$ be the probability that the stack of crates is exactly $50\mathrm{ft}$ tall, where $m$ and $n$ are coprime positive integers. Find $m$.

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