
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 a graph paper with a Cartesian coordinate plane, it is folded once such that point $A(0,2)$ coincides with point $B(4,0)$, and point $C(7,3)$ coincides with point $D(m,n)$. Find the values of $m$ and $n$, and the area $S$ of the quadrilateral $ABCD$.

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