
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
Bretschneider's theorem (the cosine theorem for a quadrilateral). Let \(a, b, c, d\) be the consecutive side lengths of a quadrilateral, \(m\) and \(n\) the lengths of its diagonals, and \(\hat{A}\) and \(\hat{C}\) the measures of two opposite angles. Then the following relation holds:

\[ 
m^{2} n^{2}=a^{2} c^{2}+b^{2} d^{2}-2 a b c d \cos (\hat{A}+\hat{C})
\]

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