
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 a quadratic function $f(x) = ax^2 + bx + c$ ($a > 0$),
(I) (i) If $b = -2$, and $f(x)$ is monotonically increasing on the interval $(1, +\infty)$, find the range of real numbers for $a$.
(ii) If $b = -1$, $c = 1$, and $|f(x)|$ has a maximum value of 1 when $x \in [0,1]$, find the range of real numbers for $a$.
(II) If $f(0) \geq 1$, $f(1) \geq 1$, and $f(x) = 0$ has two unequal positive roots that are both less than 1, find the minimum positive integer value for $a$.

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