
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 $f(x+y)-f(y)=x(x+2y+1)$ 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 $f(x)$.

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