
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 the real number \( a \) and the function \( f(x) \) be such that \( f(1) = 1 \), and for any \( x \in \mathbf{R} \backslash \{0\} \), there is \( f\left(\frac{1}{x}\right) = a f(x) - x \). Define \( A = \{ x \mid x \in \mathbf{R} \backslash \{0\}, f(x) \leqslant 1 \} \). Then, the range of the function \( g(x) = f(a x) f\left( \frac{1}{x} \right) \) (for \( x \in A \)) is __.

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