
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
Let $f, g : \mathbb{R} \to \mathbb{R}$ be functions such that
\[f(f(x) + y) = g(x) + f(x+y) + xy - x^2 + 1\] for all real numbers $x$ and $y.$
Find the expression for $f(x)$ if it is known that $g(x) = 2x + 1$ and $f(0) = 1$.

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