
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) Calculate: $$i^{2010}+(\sqrt{2}+\sqrt{2}i)^{2}-\left(\frac{\sqrt{2}}{1-i}\right)^{4}$$  
(2) Given that the function $f(x)$ satisfies $$f(x)=f'(1)e^{x-1}-f(0)x+\frac{1}{2}x^{2}$$; find the explicit formula for $f(x)$.

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