
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
Pentagon ABCDE has a vertical line of symmetry. The lengths AE and BD are each increased to 5 units, and the total area of the pentagon is now 50 square units. Determine the $y$-coordinate of vertex C. [asy]
unitsize(2mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), B=(0,5), C=(2.5, y_c), D=(5,5), E=(5,0);
pair[] dots={A,B,C,D,E};

draw(B--C--D--E--A--B);
dot(dots);

axes(Arrows(4));

label("A(0,0)",A,SW);
label("E(5,0)",E,SE);
label("D(5,5)",D,NE);
label("C",C,NE);
label("B(0,5)",B,NW);
[/asy]

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