
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 the function $f(x)=\begin{cases} \log_{a}x, & 0 < x\leqslant 1 \\ (3a-1)x+\frac{1}{2}a, & x > 1 \end{cases}$, for any $x_{1}, x_{2}$ in the domain, if $x_{1} < x_{2}$, then $f(x_{1}) > f(x_{2})$ holds. The range of the real number $a$ is ______.

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