
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 $f(x)$ such that for all real numbers $x$ and $y$, the equation $f(x+y) - f(y) = x(x+2y+1)$ holds, and $f(1) = 0$.
1. Find the value of $f(0)$.
2. Find the expression for $f(x)$.
3. Let $a\in\mathbb{R}$, and define two properties P and Q as follows: Property P: for $0 < x < \frac{1}{2}$, the inequality $f(x) + 3 < 2x + a$ always holds; Property Q: when $x \in [-2, 2]$, the function $g(x) = f(x) - ax$ is monotonic. Denote the set of all $a$ satisfying property P as $A$, and the set of all $a$ satisfying property Q as $B$. Find $A \cap \neg_\mathbb{R} B$ (where $\mathbb{R}$ is the universal set).

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