
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 domain of the function $f(x)$ is $R$, $f(2x+2)$ is an even function, $f(x+1)$ is an odd function, and when $x\in [0,1]$, $f(x)=ax+b$. If $f(4)=1$, find the value of $\sum_{i=1}^3f(i+\frac{1}{2})$.

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