
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 $(G, *)$ be a finite group, $H \subseteq G$ a subgroup of $G$, and $x, y$ two elements of $G$. Show the equivalence between the following propositions:

1. $x * H = y * H$
2. $(x * H) \cap (y * H) \neq \emptyset$
3. $x^{-1} * y \in H$

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