
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 have dimensions $3\text{ft} \times 4\text{ft} \times 6\text{ft}$. Each crate can be placed in one of three orientations. These crates need to be stacked on top of each other such that the stack of crates reaches exactly 50 feet in height. Each crate’s orientation is decided at random. Let $\frac{m}{n}$ be the probability that the stack of crates is exactly 50 feet tall, where $m$ and $n$ are relatively prime positive integers. Find $m$.

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