
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 $f(x) = a \ln x + b$ and $g(x) = x^2 + kx + 3$, the tangent line equation of the curve $y = f(x)$ at the point $(1, f(1))$ is $y = x - 1$.

1. Find the minimum value of $f(x)$ on the interval $[m, n]$ where $0 < m < n$.
2. If there exists an $x \in (\frac{1}{e}, e)$ such that the inequality $2f(x) + g(x) \geq 0$ holds, find the range of possible values for $k$.

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