
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
Let  $\mathcal{F}$  be the set of continuous functions  $f: \mathbb{R} \to \mathbb{R}$  such that   $$ e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R} $$  For  $f \in \mathcal{F},$  let   $$ I(f)=\int_0^ef(x) dx $$  Determine  $\min_{f \in \mathcal{F}}I(f).$ *Liviu Vlaicu*

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