
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
Consider the function $f(x) = \frac{ax^2 - 2}{bx + c}$, where $a, b, c \in \mathbb{Z}$ and $f(x)$ is an odd function.
1. Given $f(1) = 1$ and $f(2) - 4 > 0$, find $f(x)$.
2. If $b = 1$ and $f(x) > 1$ for all $x \in (1, +\infty)$, determine the minimum value of $a$.

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