
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 $f(x) = (x - k)e^x$ where $k \in \mathbb{R}$,
1. Find the intervals of monotonicity and the extreme values of $f(x)$.
2. Find the minimum value of $f(x)$ on the interval $x \in [1,2]$.
3. Let $g(x) = f(x) + f'(x)$. If for all $k \in \left[\frac{3}{2}, \frac{5}{2}\right]$ and for all $x \in [0,1]$ it holds that $g(x) \geq \lambda$ always, find the range of the real number $\lambda$.

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