
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 $f(x)$ and its derivative $f'(x)$ have domains of all real numbers, for any $x, y \in R$, it is always true that $f(x+y) + f(x-y) = 2f(x) \cdot f(y)$. Then $f'(x)$ must be a ______ function (fill in "odd, even, neither odd nor even"); if $f(1) = \frac{1}{2}$, then $\sum_{i=1}^{2027}f(i) = \_\_\_\_\_\_$.

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