
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 $f(x) = x^2 + bx - a\ln x$ (where $a \neq 0$),  
(1) Discuss the monotonicity of the function $f(x)$ when $b=0$;  
(2) If $x=2$ is an extremum of the function $f(x)$ and $1$ is a zero of the function $f(x)$, find the value of $a + b$;  
(3) If for any $b \in [-2, -1]$, there exists $x \in (1, e)$ such that $f(x) < 0$, find the range of the real number $a$.

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