
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 $f(x) = e^x - a(x+1)$.
1. If $a > 0$ and $f(x) \geq 0$ holds for all real numbers $x$, find the maximum value of $a$.
2. Let $g(x) = f(x) + \frac{a}{e^x}$, and let $A(x_1, y_1), B(x_2, y_2)$ $(x_1 \neq x_2)$ be any two points on the curve $y=g(x)$. If the slope of line AB is always greater than a constant $m$ for any $a \leq -1$, find the range of possible values for $m$.
3. Does there exist a positive integer $a$ such that the inequality $1^n + 3^n + \ldots + (2n-1)^n < \frac{\sqrt{e}}{e-1}(an)^n$ holds for all positive integers $n$? If it exists, find the minimum value of $a$; if not, explain why.

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