
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 parallelogram $ABCD$ as shown in the diagram. A point $E$ moves along side $AB$ and a point $G$ moves along side $CD$ such that the line segment $EG$ is parallel to $AD$. When the intersection point $F$ of segments $EG$ and $BD$ is one-fifth of the diagonal $BD$ (closer to point $D$), the area of the shaded part of the parallelogram is $1 \text{ cm}^2$ greater than when it is at two-fifths (also closer to $D$). Determine the area of the parallelogram $ABCD$.

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