
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]


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Given the functions $y=\sin x+\cos x$ and $y=2 \sqrt {2}\sin x\cos x$, determine the correct conclusion(s) from the following options:

$①$ Both functions' graphs are centrally symmetric about the point $(-\frac{π}{4},0)$;

$②$ Both functions' graphs are axially symmetric about the line $x=-\frac{π}{4}$;

$③$ Both functions are monotonically increasing in the interval $(-\frac{π}{4}, \frac{π}{4})$;

$④$ Both functions have the same minimum positive period;

$⑤$ Both functions have the same maximum value.

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