
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
Given a set of sample data $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ ($x_{1} \lt x_{2} \lt \ldots \lt x_{n}$), there is a new set of data $\frac{{x}_{1}+{x}_{2}}{2}$, $\frac{{x}_{2}+{x}_{3}}{2}$, $\ldots$, $\frac{{x}_{n-1}+{x}_{n}}{2}$, $\frac{{x}_{n}+{x}_{1}}{2}$. In comparison to the original sample data, there are four judgments for the new data: ① the mean remains the same; ② the median remains the same; ③ the range decreases; ④ the variance decreases. The correct sequence numbers for all the correct judgments are ______.

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