
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 $f(x) = a^{x} + x^{2} - x\ln a$ where $a > 0$ and $a \neq 1$.
1. Find the equation of the tangent line to the function $f(x)$ at the point $(0, f(0))$.
2. Determine the interval where the function $f(x)$ is monotonically increasing.
3. If there exist $x_{1}$ and $x_{2} \in [-1, 1]$ such that $|f(x_{1}) - f(x_{2})| \geq e - 1$ (where $e$ is the base of the natural logarithm), find the range of real numbers for $a$.

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