
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 $f(x)=\begin{cases} (1-2a)^{x}, & x\leqslant 1 \\ \log_{a}x+\frac{1}{3}, & x > 1 \end{cases}$, when $x_{1} \ne x_{2}$, determine the range of the real number $a$ for which the expression $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0$.

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