
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 additive group  $\mathbb{Z}^{2}$ . Let  $H$  be the smallest subgroup containing  $(3,8), (4,-1)$  and  $(5,4)$ .
Let  $H_{xy}$  be the smallest subgroup containing  $(0,x)$  and  $(1,y)$ . Find some pair  $(x,y)$  with  $x>0$  such that  $H=H_{xy}$ .

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