
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
Let $f(x) = 2x^2 + bx - a\ln x$.
(1) When $a=5$ and $b=-1$, find the monotonic intervals of $f(x)$.
(2) If for any $b \in [-3, -2]$, there exists $x \in (1, e^2)$ (where $e$ is the base of the natural logarithm) such that $f(x) < 0$, find the range of real numbers for $a$.

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