
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 \( A \) be a set of \( m \) real numbers with the sum of its elements equal to \( a \); \( B \) be a set of \( n \) real numbers with the sum of its elements equal to \( b \). Define
\[ 
C = \{\boldsymbol{x} \mid \boldsymbol{x} = (s, t), s \in A, t \in B\}. 
\]

Let \(\boldsymbol{x}\) and \(\boldsymbol{y}\) each range over the vectors in \( C \) (allowing \(\boldsymbol{x} = \boldsymbol{y}\)) and compute \( x \cdot y \). Then the sum of these dot products \(\sum_{x y \in C} x \cdot y\) equals (express the result in terms of \( a, b, m, n \)).

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