
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
Xiao Ming's math research homework sheet has the following question: Given $-x+y=2$, and $x \lt 3$, $y\geqslant 0$, let $w=x+y-2$, then what is the range of values for $w$? <br/>【Review】<br/>Xiao Ming recalls a similar simple question he has done before: Given: $-1 \lt x \lt 2$, let $y=x+1$, then the range of values for $y$ is ______. (Please write down the answer directly) <br/>【Exploration】<br/>Xiao Ming thinks: he can convert the complex problem on the research sheet into a similar question as reviewed above. From $-x+y=2$, we get $y=2+x$, then $w=x+y-2=x+2+x-2=2x$, from $x \lt 3$, $y\geqslant 0$, we get the one-variable linear inequality group about $x$ as ______, solving this inequality group we get the range of values for $x$ as ______, then the range of values for $w$ is ______. <br/>【Application】<br/>$(1)$ Given $a-b=4$, and $a \gt 1$, $b \lt 2$, let $t=a+b$, find the range of values for $t$; <br/>$(2)$ Given $a-b=n$ ($n$ is a constant greater than $0$), and $a \gt 1$, $b\leqslant 1$, the maximum value of $2a+b$ is ______ (expressed in an algebraic expression containing $n$); <br/>【Extension】<br/>If $3x=6y+12=2z$, and $x \gt 0$, $y\geqslant -4$, $z\leqslant 9$, let $m=2x-2y-z$, and $m$ is an integer, then the sum of all possible values of $m$ is ______.

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