
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 that the function $f(x)$ satisfies the equation $f(x+y)-f(y)=x(x+2y+1)$ for all real numbers $x,y$, and $f(1)=0$.

(1) Find the value of $f(0)$.
(2) Find the analytic expression of $f(x)$.
(3) Given $a\in R$, let $P$: the inequality $f(x)+3 < 2x+a$ holds when $0 < x < \frac{1}{2}$; $Q$: $g(x)=f(x)-ax$ is a monotonic function when $x\in [-2,2]$. If the set of $a$ that satisfies $P$ is denoted as $A$, and the set of $a$ that satisfies $Q$ is denoted as $B$, find $A∩{{C}_{R}}B(R$ is the universal set$)$.

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