
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
A function $f(x)$ is defined on $(0,+\infty)$ and satisfies: for all $x\in(0,+\infty)$, $f(2x)=2f(x)$; when $x\in(1,2]$, $f(x)=2-x$. Determine the validity of the following conclusions:

$\text{①}$ For all $m\in\mathbb{Z}$, $f(2^{m})=0$;

$\text{②}$ The range of the function $f(x)$ is $[0,+\infty)$;

$\text{③}$ There exists an $n\in\mathbb{Z}$ such that $f(2^{n}+1)=9$;

$\text{④}$ A sufficient condition for the function $f(x)$ to be monotonically decreasing on the interval $(a,b)$ is "there exists a $k\in\mathbb{Z}$ such that $(a,b) \subseteq (2^{k},2^{k+1})$".

Please fill in the serial numbers of all correct propositions (e.g., ①③).

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