
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 $f(x)$ on $R$ such that $f(-x) = f(x)$, and for any distinct real numbers $x_{1}$, $x_{2} \in [0,+\infty)$, $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0$ holds, determine the range of real number $m$ for which the inequality $f(2mx-\ln x-3) \geq 2f(3)-f(-2mx+\ln x+3)$ always holds for $x \in [1,3]$.

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