
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 $f(x)=\cos (2x+ \frac {π}{3})+1$, choose the correct conclusion(s) from the options below. Write down the number(s) of all correct conclusions:

$①$ The point $(- \frac {5}{12}π,0)$ is a center of symmetry of the graph of the function $f(x)$;

$②$ The line $x= \frac {π}{3}$ is an axis of symmetry of the graph of the function $f(x)$;

$③$ The minimum positive period of the function $f(x)$ is $π$;

$④$ The function $f(x)$ is increasing on the interval $[- \frac {π}{6}, \frac {π}{3}]$;

$⑤$ After shifting the graph of the function $f(x)$ to the right by $\frac {π}{6}$ units, the corresponding function is an even function.

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