
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
A right rectangular prism, denoted as $Q$, has integral side lengths $a, b, c$, arranged so that $a \le b \le c$. A plane, parallel to one of its faces, divides $Q$ into two smaller prisms, one of which is similar to $Q$ and both having non-zero volumes. Provided that $b = 1536$, determine the number of ordered triples $(a, b, c)$ that allow for such a plane to exist.

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