
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 lengths $a, b, c$ 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 $Q$, and both of which have nonzero volume. Provided that $b=500$ and $a$ must be odd, determine how many ordered triples $(a, b, c)$ permit such a plane to exist.

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