
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
(1) Given the function $f(x)=13-8x+\sqrt{2}x^{2}$, and $f'(x_{0})=4$, find the value of $x_{0}$.
(2) Given the function $f(x)=x^{2}+2xf'(0)$, find the value of $f'(0)$.

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