
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 $f(x)$ satisfies $f(x+y)-f(y)=x(x+2y+1)$ for all real numbers $x,y\in\mathbb{R}$, and $f(1)=0$.  
$(1)$ Find the value of $f(0)$;  
$(2)$ Find the expression for $f(x)$;  

$(3)$ Given $a\in\mathbb{R}$, let $P$: for $0 < x < \frac{1}{2}$, the inequality $f(x)+3 < 2x+a$ always holds; $Q$: for $x\in[-2,2]$, $g(x)=f(x)-ax$ is a monotonic function. 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 $A\cap(\mathbb{R}\setminus B)$.

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