
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 \( d, e, f \) with \( d \le e \le f \). A plane parallel to one of the faces of \( Q \) cuts \( Q \) into two prisms, one of which is similar to \( Q \), with both prisms having nonzero volume. Given that \( e = 1980 \), determine the number of ordered triples \( (d, e, f) \) for which such a plane exists.

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