
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 functions $f(x) = ax^2 - \ln x$ and $g(x) = \frac{1}{2}ax^2 + x$ , where $a \in \mathbb{R}$:

1. Find the interval of increase for $f(x)$ when $a = \frac{1}{2}$.
2. Let $F(x) = f(x) - g(x)$. If the inequality $F(x) \geq 1 - ax$ always holds, find the minimum integer value of $a$.

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