
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 $f(x) = ae^x - x$ ($a \in \mathbb{R}$), where $e$ is the base of the natural logarithm, $e = 2.71828\ldots$

(Ⅰ) Determine the monotonicity of the function $f(x)$ and explain the reason.

(Ⅱ) If $x \in [1, 2]$, the inequality $f(x) \geq e^{-x}$ always holds, find the range of values for $a$.

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