
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
(1) The ratio of the volume of the circumscribed sphere to the inscribed sphere of a regular tetrahedron with edge length $\sqrt{2}$ is \_\_\_\_\_\_.

(2) The maximum value of the function $y=\cos^2 x+\sin x$ is \_\_\_\_\_\_.

(3) If the sequence ${ a\_n }$ satisfies $\frac{1}{a\_{n+1}}-\frac{1}{a\_n}=d$, ($n\in N^{*}$, $d$ is a constant), then the sequence ${ a\_n }$ is called a harmonic sequence. It is known that the sequence ${\frac{1}{x\_n}}$ is a harmonic sequence and $x\_1+x\_2+...+x\_{10}=100$, then $x\_4+x\_7=$ \_\_\_\_\_\_.

(4) Let the sum of the first $n$ terms of the sequence ${ a\_n }$ be $S\_n(n\in N^{*})$. There are the following propositions about the sequence ${ a\_n }$:
({①}) If ${ a\_n }$ is both an arithmetic sequence and a geometric sequence, then $a\_n=a\_{n+1}(n\in N^{*})$;
({②}) If $S\_n=an^2+bn$, ($a,b\in R$), then ${ a\_n }$ is an arithmetic sequence;
({③}) If $S\_n=1-(-1)^n$, then ${ a\_n }$ is a geometric sequence;
({④}) If ${ a\_n }$ is a geometric sequence, then $S\_m$, $S\_{2m}-S\_m$, $S\_{3m}-S\_{2m}(m\in N^{*})$ also form a geometric sequence;
The correct propositions are \_\_\_\_\_\_.

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