
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 $f(x+y)-f(y)=(x+2y+1)x$ for all real numbers $x$, $y$, and $f(1)=0$.

(1) Find $f(0)$;  
(2) Find $f(x)$;  
(3) The inequality $f(x) > ax - 5$ holds for $0 < x < 2$, find the range of $a$.

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