
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
Define a function $f(x)$ on $\mathbb{R}$ whose graph is symmetric about the $y$-axis, and $f(x)$ is monotonically decreasing on $[0,+\infty)$. If the inequality $f(2mx-\ln x-3) \geqslant 2f(3)-f(-2mx+\ln x+3)$ holds for all $x \in [1,3]$, determine the range of the real number $m$.

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