
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 $f(x) = x \ln x$ (where $e = 2.718\ldots$),
(1) Set $g(x) = f(x) + x^2 - 2(e+1)x + 6$,
   ① Let $g'(x)$ be the derivative of $g(x)$. Calculate $g'(e)$.
   ② If the equation $g(x) - a = 0$ has two distinct real roots, find the range of real numbers for $a$.
(2) If there exists a point $x_0$ in the interval $[1, e]$ such that $m(f(x_0) - 1) > x_0^2 + 1$, find the range of the real number $m$.

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