
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
Let $a$ and $b$ be distinct rational numbers. Suppose there exist distinct rational numbers $r$ and $s$ such that for every rational number $z$, we have \[
  (z - r)(z - s) = (z - ar)(z - bs).
\]
Compute the number of distinct possible values of $a$ and $b$.

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