
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 $f(x)=\begin{cases}(3a-1)x+4a, & (x < 1) \\ \log_a x, & (x\geqslant 1)\end{cases}$ 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