
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 $f(x) = e^x - x$ (where $e$ is the base of the natural logarithm),
1. Find the minimum value of $f(x)$;
2. The solution set of the inequality $f(x) > ax$ is $P$. If $M = \{x | \frac{1}{2} \leq x \leq 2\}$ and $M \cap P \neq \emptyset$, find the range of the real number $a$;
3. Given $n \in \mathbb{N}^*$, and $S_n = \int_{t}^{n} [f(x) + x] dx$ (where $t$ is a constant, $t \geq 0$), is there a geometric sequence $\{b_n\}$ such that $b_1 + b_2 + \ldots + b_n = S_n$? If it exists, find the general formula of the sequence $\{b_n\}$; if not, explain why.

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