
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
You are given the following propositions:

$①$ The minimum positive period of the function $y=\cos^2 x - \frac{1}{2}$ is $\pi$;

$②$ The set of terminal side angles on the $y$-axis is $\{α|α=\frac{kπ}{2},k∈\mathbb{Z}\}$;

$③$ The graph of the function $y=4\cos (2x+θ)$ is symmetrical about the point $(\frac{π}{6},0)$, hence $θ=\frac{kπ}{2}+\frac{π}{6}$ $(k∈\mathbb{Z})$;

$④$ The function $f(x)=3\sin (2x-\frac{π}{3})$ is increasing in the interval $(-\frac{π}{12},\frac{5π}{12})$;

$⑤$ The graph of the function $y=4\cos 2x$ can be obtained by shifting the graph of $y=4\sin 2x$ to the right by $\frac{π}{4}$ units.

Identify the correct proposition(s) by their number(s).

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