
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 $f(x) = e^x(\ln x + k)$, where $k$ is a constant, and $e=2.71828…$ is the base of the natural logarithm. The derivative of the function $y=f(x)$ is $f'(x)$, and $f'(1)=0$.

(1) Find the value of $k$;  
(2) Let $g(x) = f'(x) - 2[f(x) + e^x]$, $\varphi(x) = \frac{e^x}{x}$, and $g(x) \leq t \cdot \varphi(x)$ always holds. Find the range of the real number $t$.

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