
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 $f(x) = e^{ax} - x - b$.
1. When $b=1$, the function $f(x)$ has two zeros. Find the range of real number $a$.
2. Let $b=0$, the function $f(x)$ has two zeros $x_{1}$ and $x_{2}$, where $x_{1} < x_{2}$. As $a$ varies, if $\lambda \ln x_{1} + \ln x_{2}$ attains a minimum value of $e$ (where $e$ is the base of natural logarithm), find the value of the constant $\lambda$.

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