
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
Among the following judgments:  
① $x^2 \neq y^2 \Leftrightarrow x \neq y$ or $x \neq -y$;  
② If $x^2 + y^2 = 0$, then both $x$ and $y$ are zero;  
③ The proposition "$\emptyset \subseteq \{1, 2\}$ or $-1 \in \mathbb{N}$" is true;  
④ "$am^2 < bm^2$" is a necessary and sufficient condition for "$a < b$";  
⑤ If $b \leq -1$, then the equation $x^2 - 2bx + b^2 + b = 0$ has real roots.  
Among these, the correct ones are (fill in the numbers).

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