
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 $f(x) = a\ln x + x^2$ (where $a$ is a real constant).  
(1) When $a = -4$, find the maximum value of the function $f(x)$ on the interval $[1, e]$ and the corresponding value of $x$;  
(2) Discuss the number of roots of the equation $f(x) = 0$ when $x \in [1, e]$.

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