
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 \( f(x) \) be a function defined for all real numbers \( x \neq 0 \) such that for any non-zero real numbers \( x_1 \) and \( x_2 \), the equation \( f(x_1 x_2) = f(x_1) + f(x_2) \) holds. Additionally, \( f(x) \) is an increasing function on the interval \( (0, +\infty) \). Determine the solution set for the inequality \( f(x) + f\left(x - \frac{1}{2}\right) \leqslant 0 \).

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