
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
Let the function $f(x) = \frac{a+1}{2}x^2 - ax (a \in \mathbb{R})$, $g(x) = \ln x$, $F(x) = f(x) - g(x)$.

1. If $a = -3$, $x \in [\frac{1}{e^2}, e^2]$, find the maximum and minimum values of $F(x)$.
2. If $\forall a \in (-3, -2)$ and $\forall x_1, x_2 \in [1, 2]$, there always exists $|F(x_1) - F(x_2)| < at + \ln 2$, find the range of the real number $t$.

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