
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 $A = \{1, 2, 3, \ldots, 2n\}$ (where $n \in \mathbb{N}^*$), for a subset $S$ of $A$, if $S$ satisfies property $P$: "There exists a positive integer $m \leq n$ such that for any pair of elements $s_1, s_2$ in $S$, $|s_1 - s_2| \neq m$," then $S$ is called an ideal set. Consider the following statements:
- ① When $n=10$, the set $B=\{x \in A | x > 9\}$ is an ideal set;
- ② When $n=10$, the set $C=\{x \in A | x = 3k - 1, k \in \mathbb{N}^*\}$ is an ideal set;
- ③ When $n=1,000$, if the set $S$ is an ideal set, then the set $T=\{2,001 - x | x \in S\}$ is also an ideal set.

Identify which of the propositions are true (list all the true statement numbers).

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