
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 the function $y=f(x)$ that satisfies $f(-x)=-f(x)$ and $f(x+1)=-f(x)$, determine the correct statement(s) from the options below.

$①$ The function $y=f(x)$ is periodic with a period of $2$.

$② f(x)=\sin \pi x$ is a function that satisfies the given conditions.

$③$ The center of symmetry of the function $y=f(x)$ is $(k,0)$.

$④$ The interval of monotonic increase of the function is $\left[- \frac{1}{2}+2k,\frac{1}{2}+2k \right]$.

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