
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 following propositions:  
① The solution set of the inequality $(a-2)x^2+(a-2)x+1>0$ with respect to $x$ is $\mathbb{R}$ if and only if $2<a<6$;  
② We define a "grandchild set" of a non-empty set $A$ as a proper subset of a proper subset of $A$, then the set $\{1,3,5,7,9\}$ has 26 "grandchild sets".  
③ Given $f(x)=ax^2+bx+c$ ($a\neq 0$), if the equation $f(x)$ has no real roots, then the equation $f[f(x)]=x$ also has no real roots;  
④ If $\{a_n\}$ is a geometric sequence, and $S_n$ is the sum of the first $n$ terms, then $S_4$, $S_8-S_4$, $S_{12}-S_8$ form a geometric sequence.  
The correct propositions are numbered as follows.

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