
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
In the arithmetic sequence \(\{a_{n}\}\), \(a_{20}=\frac{1}{a}, a_{201}=\frac{1}{b}, a_{2012}=\frac{1}{c}\). Find the value of \(1992 a c - 1811 b c - 181 a b\).

( Note: Given constants \(a\) and \(b\) satisfy \(a, b > 0, a \neq 1\), and points \(P(a, b)\) and \(Q(b, a)\) are on the curve \(y=\cos(x+c)\), where \(c\) is a constant. Find the value of \(\log _{a} b\).

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