
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:

$$
\left\{
\begin{array}{l}
\gcd(a, b, c) = 10 \\
\operatorname{lcm}(a, b, c) = 2^{17} \cdot 5^{16}
\end{array}
\right.
$$

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