
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
In the pentagon ABCDE with a vertical line of symmetry, if the area of the square $ABDE$ becomes 10 square units, what must be the $y$-coordinate of vertex C so that the area of the triangle $BCD$ is 30 square units? Assume that A(0,0), B(0,4), D(4,4), and E(4,0).

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