
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 $f(x) = a\ln x - bx^2$, where $a, b \in \mathbb{R}$. If the inequality $f(x) \ge x$ holds for all $b \in (-\infty, 0]$ and $x \in (e, e^2]$, determine the range of values for $a$.

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