
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
Another trapezoid \(ABCD\) has \(AD\) parallel to \(BC\). \(AC\) and \(BD\) intersect at \(P\). If \(\frac{[ADP]}{[BCP]} = \frac{1}{2}\), find \(\frac{[ADP]}{[ABCD]}\). (Here, the notation \(\left[P_1 \cdots P_n\right]\) denotes the area of the polygon \(P_1 \cdots P_n\)).

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