
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
Sequence $A_{n}$: $a_{1}$, $a_{2}$, $\ldots a_{n}(n\geqslant 4)$ satisfies: $a_{1}=1$, $a_{n}=m$, $a_{k+1}-a_{k}=0$ or $1$. For any $i$, $j$, there exist $s$, $t$ such that $a_{i}+a_{j}=a_{s}+a_{t}$, where $i$, $j$, $s$, $t\in \{1,2,\ldots ,n\}$ and are all distinct.<br/>$(1)$ When $m=2$, write down the sequence numbers that meet the conditions in the following three sequences: ① $1$, $1$, $1$, $2$, $2$, $2$; ② $1$, $1$, $1$, $1$, $2$, $2$, $2$, $2$; ③ $1$, $1$, $1$, $1$, $1$, $2$, $2$, $2$, $2$, $2$;<br/>$(2)$ Let $S=a_{1}+a_{2}+\ldots +a_{n}$. If $m=3$, find the minimum value of $S$;<br/>$(3)$ If $m=1000$, find the minimum value of $n$.

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