
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 $f(x) = x\ln x - \alpha x + 1$ has an extremum at $x=e^{2}$.  
$(1)$ Find the intervals where $f(x)$ is monotonic.  
$(2)$ If $f(x) < 2c^{2} - c$ holds for all $x \in [1, e^{3}]$, find the range of real number $c$.

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