
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 $f(x) = e^x + 2ax$.

(Ⅰ) Find the intervals of monotonicity for the function $f(x)$;

(Ⅱ) If the minimum value of the function $f(x)$ on the interval $[1, +\infty)$ is $0$, find the value of $a$;

(Ⅲ) If for any $x \geqslant 0$, $f(x) \geqslant e^{-x}$ always holds, find the range of values for $a$.

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