
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 $$f(x)= \begin{cases} \left.\begin{matrix}(3a-1)x+4a，x＜1 \\ \log_{a}x，x≥1\end{matrix}\right.\end{cases}$$ it satisfies: for any real numbers $x_1$, $x_2$, when $x_1＜x_2$, it always holds that $f(x_1)-f(x_2)＞0$. Determine the range of the real number $a$.

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