
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 $f(x)=|\sin x|+\cos x$, there are several propositions as follows:

$①$ The function is an even function;

$②$ The smallest positive period of the function is $\dfrac{\pi}{2}$;

$③$ The range of the function is $[-1,\sqrt{2}]$;

$④$ If the length of the interval $(a,b)$ is defined as $b-a$, then the maximum length of the interval where the function is monotonically increasing is $\dfrac{3\pi}{4}$.

Among these propositions, the correct ones are ______.

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