
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 below, three are true and one is false:
1) \(a^{2} + 4a + 3\) is divisible by \(b\);
2) \(a^{2} + ab - 6b^{2} - 2a - 16b - 8 = 0\);
3) \(a + 2b + 1\) is divisible by 4;
4) \(a + 6b + 1\) is a prime number.

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