
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 $f(x)=a\ln x+x^{2}$ $(a$ is a real constant$)$.
(1) When $a=-4$, find the maximum value of the function $f(x)$ on $[1,e]$ and the corresponding $x$ value;
(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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