
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
For the inequality system about $x$: $\left\{\begin{array}{l}{x-a≥1}\\{x-b＜2}\end{array}\right.$.<br/>①If the solution set of the inequality system is $-1\leqslant x \lt 3$, then $a=-2$, $b=1$;<br/>②If $a=b$, then the solution set of the inequality system is $a+1\leqslant x \lt a+2$;<br/>③If the inequality system has no solution, then $a \gt b+1$;<br/>④If the inequality $x-a\geqslant 1$ has only $5$ negative integer solutions, then $-7 \lt a\leqslant -6$.<br/>Which of the following statements is correct?

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