
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
Find all pairs of natural numbers \( a \) and \( b \) such that out of the four statements:

5) \( a^2 + 6a + 8 \) is divisible by \( b \);

6) \( a^2 + ab - 6b^2 - 15b - 9 = 0 \);

7) \( a + 2b + 2 \) is divisible by 4;

8) \( a + 6b + 2 \) is a prime number;

three are true, and one is false.

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