
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 \( a \) be a given constant with \( 0 < a < 1 \). Suppose \( f(x) \) is an odd function on \( \mathbf{R} \) and is decreasing on \( (0, +\infty) \). If \( f\left(\frac{1}{2}\right) = 0 \) and \( f\left(\log_{a} x\right) > 0 \), then the range of \( x \) is \(\qquad\).

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