
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 \(a\) and \(b\) are integers, what is the smallest positive integer value \(c\) for \(\Delta=\left|\begin{array}{ll}36 & a \\ 81 & b\end{array}\right|\)? Also, among all pairs of positive integer solutions \((a, b)\) that satisfy \(\Delta=c\), how can \(a\) and \(b\) be chosen such that \(a+b\) is minimized?

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