
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 function \( f(x) \) (for \( x \in \mathbf{R}, x \neq 0 \)) satisfy the condition \( f(x_{1} x_{2}) = f(x_{1}) + f(x_{2}) \) for any nonzero real numbers \( x_{1} \) and \( x_{2} \), and \( f(x) \) is an increasing function over \( (0, +\infty) \). Then the solution to the inequality \( f(x) + f\left(x - \frac{1}{2}\right) \leqslant 0 \) is ______.

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