
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 $f(x) = mx^2 + (1-3m)x + 2m - 1$.
- (I) Let $m=2$, and the solution set of $f(x) \leq 0$ is $A$. Set $B = (a, 2a+1]$ ($a > 0$). If $A \subseteq B$, find the range of $a$.
- (II) Find the solution set $S$ of the inequality $f(x) \leq 0$ with respect to $x$.
- (III) If there exists $x > 0$ such that $f(x) > -3mx + m - 1$, find the range of the real number $m$.

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