
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 \(Q\) has sides of integral length \(a, b, c,\) with \(a \leq b \leq c\). A plane parallel to one of the faces of \(Q\) cuts \(Q\) into two prisms, one of which is similar to \(Q\), each having non-zero volume. Given that \(b = 2023\), find the number of ordered triples \((a, b, c)\) for which such a plane exists.

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