
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 $x, y$ be any real numbers, and define the operation: $x*y = (x+1)(y+1) - 1$. The following five propositions are obtained:  
① $x*y = y*x$; ② $x*(y+z) = x*y + x*z$; ③ $(x+1)*(x-1) = (x*x) - 1$; ④ $x*0 = 0$; ⑤ $(x+1)*(x+1) = x*x + 2*x + 1$;  
Among these, the correct propositions are numbered as follows.

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