
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 $a$ and $b$ are integers, determine the smallest positive integer value \( c \) of the determinant \(\Delta = \left|\begin{array}{ll}36 & b \\ 81 & a\end{array}\right|\). For the pairs of positive integers \((a, b)\) that satisfy \(\Delta = c\), find the values of \( a \) and \( b \) such that \( a + b \) is minimized. What are the values of \( a \) and \( b \)?

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