
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 the following propositions:  
① The negation of the proposition "If $x^2=1$, then $x=1$" is "If $x^2=1$, then $x\neq1$";  
② "$x=-1$" is a necessary but not sufficient condition for "$x^2-5x-6=0$";  
③ The negation of the proposition "There exists an $x$ in $\mathbb{R}$ such that $x^2+x-1<0$" is: "For all $x$ in $\mathbb{R}$, $x^2+x-1>0$";  
④ The contrapositive of the proposition "If $x=y$, then $\sin x=\sin y$" is true.  
Among these, the correct proposition numbers are $\boxed{\text{④}}$.

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