
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
The function $f(x)$ satisfies $f(x+y) - f(y) = (x+2y+2)x$ for all real numbers $x$ and $y$, and $f(2) = 12$.

1. Find the value of $f(0)$.
2. There exists an $x_0 \in \mathbb{R}$ in the interval $(1, 4)$ such that $f(x_0) - 8 = ax_0$ holds true. Determine the range of the real number $a$.

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