
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
Given that \( m, n, k \) are positive integers, if there exists a pair of positive integers \( (a, b) \) such that
\[
(1+a) n^{2}-4(m+a) n+4 m^{2}+4 a+b(k-1)^{2}<3,
\]
then the number of possible values of \( m+n+k \) is \(\quad\) .

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