
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
Let $S$ be a set containing at least two elements, and a binary operation "$*$" is defined on $S$ (i.e., for any $a, b \in S$, for the ordered pair $(a, b)$, there is a unique element $a*b$ in $S$ corresponding to it). If for any $a, b \in S$, it holds that $a*(b*a) = b$, then for any $a, b \in S$, consider the following relations: ① $(a*b)*a = a$; ② $[a*(b*a)]*(a*b) = a$; ③ $b*(b*b) = b$; ④ $(a*b)*[b*(a*b)] = b$. Identify the correct statements among these (Write down the numbers of all correct statements).

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