
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
Let the domain of the function $f(x)$ be $D$. If there exists a non-zero real number $l$ such that for any $x \in M$ ($M \subseteq D$), we have $x+l \in D$, and $f(x+l) \geq f(x)$, then $f(x)$ is called an "$l$-increasing function" on $M$. Consider the following propositions:  
① The function $f(x) = \log_{2}^{x}$ is a "1-increasing function" on $(0, +\infty)$;  
② The function $f(x) = \cos 2x$ is a "$\pi$-increasing function" on $\mathbb{R}$;  
③ If the function $f(x) = x^{2}$, whose domain is $[-1, +\infty)$, is an "$m$-increasing function" on $[-1, +\infty)$, then the range of the real number $m$ is $[-1, +\infty)$.  
Among these propositions, the correct ones are $\boxed{\text{①②}}$.

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