
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 $P$ has sides of integral length $a, b, c,$ with $a \le b \le c$. A plane cuts $P$ from one vertex to the opposite vertex on a diagonal slice, resulting in two regions, one of which is a smaller, similar prism $P'$, each having nonzero volume. Given $b = 1995$, for how many ordered triples $(a, b, c)$ does such a plane exist?

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