
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 $f(x) = ax^2 - 2x + 2$ where $a \in \mathbb{R}$,
1. If $h(10^x) = f(x) + x + 1$, find the explicit form of $h(x)$.
2. If $f(x) > 0$ holds true for all $x \in [1, 2]$, determine the range of values for $a$.
3. Let the function $F(x) = |f(x)|$. If for any $x_1, x_2 \in [1, 2]$ where $x_1 \neq x_2$, it satisfies $\frac{F(x_1) - F(x_2)}{x_1 - x_2} > 0$, find the range of values for the real number $a$.

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