
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
Let the domain of the function $f(x)$ be $R$, and $f(x)$ satisfies the following properties: $\left(i\right)$ If the graph of $f(x)$ is shifted $2$ units to the left, the resulting graph is symmetric about the $y$-axis, $\left(ii\right)$ If all the points on the graph of $f(x)$ have their y-coordinates unchanged, the x-coordinates are shortened to $\frac{1}{2}$ of the original, and then shifted $ \frac{1}{2} $ units to the left, the resulting graph is symmetric about the origin. Given the following four conclusions: <br/>① $f(1) = f(3)$; <br/>② $f(0) = 0$; <br/>③ $f(2) + f(4) = 0$; <br/>④ $f(-\frac{1}{2})f(\frac{11}{2}) \leq 0$. The correct conclusions are ______.

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