
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
Consider the function \( g(x) = \frac{px+q}{rx+s} \), where \( p, q, r, \) and \( s \) are nonzero real numbers, with the properties that \( g(3) = 3 \), \( g(81) = 81 \), and \( g(g(x)) = x \) for all values except \( \frac{-s}{r} \). Determine the unique number that is not in the range of \( g \).

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