
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 four propositions:
1. The converse of "If $xy=1$, then $x$ and $y$ are reciprocals of each other."
2. The negation of "Congruent triangles have equal perimeters."
3. The contrapositive of "If $b \leq -1$, then the equation $x^2 - 2bx + b^2 + b = 0$ has real roots."
4. There exist $\alpha$ and $\beta$ in $\mathbb{R}$ such that $\sin(\alpha + \beta) = \sin \alpha + \sin \beta$.
Identify which propositions are true.

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