
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 points $A = (0,0)$, $B = (1,2)$, $C = (3,0)$, and $D = (2,2)$. If a line passing through $A$ divides trapezoid $ABCD$ into two regions of equal area and intersects line $\overline{BD}$, determine the coordinates $(\frac{p}{q}, \frac{r}{s})$ of the intersection point in their lowest terms.

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