
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 $f(x)$ for all $x, y \in \mathbb{R}$ satisfies $f(x+y) - f(y) = x(x+2y+1)$, and $f(1)=0$.
1. Find the value of $f(0)$;
2. Find the analytic expression of $f(x)$;
3. Let $a \in \mathbb{R}$, define P as the semi-open interval where $0 \leq x \leq \frac{3}{4}$ such that the inequality $f(x) + 3 < 2x + a$ holds, and define Q as the condition where $x\in [-2, 2]$ and $g(x) = f(x) - ax$ is a monotonic function. Denote the set of all real numbers $a$ that satisfy condition P as $A$, and those that satisfy condition Q as $B$. Find $A \cap \mathbb{C}_\mathbb{R} B$, where $\mathbb{C}_\mathbb{R} B$ is the complement of $B$ in $\mathbb{R}$.

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