
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 $f(x)={\log_{\frac{1}{3}}}({{x^2}-ax-a})$ satisfies the inequality $\frac{{f({{x_2}})-f({{x_1}})}}{{{x_2}-{x_1}}} \gt 0$ for any two distinct real numbers $x_{1}$, $x_{2}\in (-\infty $, $-\frac{1}{2}$), the range of real number $a$ is ______.

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