
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
If the function $G$ has a maximum value of $M$ and a minimum value of $N$ on $m\leqslant x\leqslant n\left(m \lt n\right)$, and satisfies $M-N=2$, then the function is called the "range function" on $m\leqslant x\leqslant n$. <br/>$(1)$ Functions ① $y=2x-1$; ② $y=x^{2}$, of which function ______ is the "range function" on $1\leqslant x\leqslant 2$; (Fill in the number) <br/>$(2)$ Given the function $G:y=ax^{2}-4ax+3a\left(a \gt 0\right)$. <br/>① When $a=1$, the function $G$ is the "range function" on $t\leqslant x\leqslant t+1$, find the value of $t$; <br/>② If the function $G$ is the "range function" on $m+2\leqslant x\leqslant 2m+1(m$ is an integer), and $\frac{M}{N}$ is an integer, find the value of $a$.

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