
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
Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations

\[
\begin{cases}
\gcd(a, b, c) = 22 \\
\mathrm{lcm}(a, b, c) = 2^{16} \cdot 11^{19}
\end{cases}
\]

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