
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
Consider the binomial expression $(x-1)^{2009}$. The following statements are made about it:  
① The sum of all coefficients of non-constant terms in this binomial is 1.  
② The fifth term of the expanded expression is $-C^{5}_{2009}x^{2004}$.  
③ The term with the highest coefficient in this binomial is the 1005th term.  
④ When $x=2009$, the remainder of $(x-1)^{2009}$ divided by 2009 is 2008.  
Among these statements, the correct ones are _____ (fill in the correct numbers).

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