
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
Determine the explicit formulas for the following functions based on the given conditions:
(1) Given that $f(x)$ is a quadratic function, and $f(0)=0$, $f(x+1)=f(x)+x+1$, find $f(x)$.
(2) Given that $f(\sqrt{x}+1)=x+2\sqrt{x}$, find $f(x)$.
(3) If $f(x)$ satisfies $f(x)+2f\left(\frac{1}{x}\right)=ax$, find $f(x)$.

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