
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 of dimensions $3\mathrm{ft} \times 5\mathrm{ft} \times 7\mathrm{ft}$. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, with the orientation chosen at random. What is the probability that the stack of crates reaches exactly $43\mathrm{ft}$ tall? Represent the probability as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find the value of $m$.

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