
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 nonzero real numbers \(a, b, c, d\) and the function \(f(x)=\frac{ax+b}{cx+d}\) for \(x \in \mathbb{R}\) such that \(f(19) = 19\) and \(f(97) = 97\). If for any real number \(x \neq -\frac{d}{c}\), it holds that \(f[f(x)] = x\), find the unique number that is outside the range of \(f(x)\).

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