
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
Let the points \( A = 3 + 2i \) and \( B = -1 + 4i \) define a line, and the midpoint of the line segment joining two other complex numbers \( C = 2 + 3i \) and \( D = 4 - i \). Find the complex numbers \( a \) and \( b \) such that the line can be expressed in the form
\[ az + b \overline{z} = c \]
and determine the value of \( ab \).

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