
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
Given that \(a, b, c, d\) are non-zero real numbers and \(f(x)=\frac{a x+b}{c x+d}\), where \(x \in \mathbf{R}\), and it is known that \(f(19) = 19\) and \(f(97) = 97\). If for any real number \(x\) (where \(x \neq -\frac{d}{c}\)), \(f[f(x)] = x\) holds true, find the only number outside the range of \(f(x)\).

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