
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 the group \( \{ (m, n) : m, n \text{ are integers} \} \) with the operation \( (a, b) + (c, d) = (a + c, b + d) \). Let \( H \) be the smallest subgroup containing \( (3, 8) \), \( (4, -1) \) and \( (5, 4) \). Let \( H_{ab} \) be the smallest subgroup containing \( (0, a) \) and \( (1, b) \). Find \( a > 0 \) such that \( H_{ab} = H \).

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