
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 functions $f(x) = (x + a) \ln x$ and $g(x) = \frac{2x^2}{e^x}$. The tangent line of the curve $y = f(x)$ at $x = 1$ passes through the point $(2, 3)$.

(1) Find the value of the real number $a$.
(2) Determine if there exists a natural number $k$ such that the function $y = f(x) - g(x)$ has a unique zero in the interval $(k, k + 1)$. If it exists, find $k$; if not, explain the reason.
(3) Let $h(x) = \min\{f(x), g(x)\}$. For a real number $m$, there exists $x_0 \in (0, +\infty)$ such that $h(x_0) \geq m$ holds. Find the range of values for $m$.

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