
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
Determine the length $y$ in the following quadrilateral given the lengths of the sides and one diagonal:
[asy]
unitsize(0.7 cm);

pair A, B, C, D, O;

O = (0,0);
A = 4*dir(140);
B = 5*dir(140 + 180);
C = 8*dir(40);
D = 6*dir(40 + 180);

draw(A--B);
draw(C--D);
draw(A--C);
draw(B--D);

label("$6$", (A + O)/2, SW);
label("$11$", (C + O)/2, SE);
label("$6$", (D + O)/2, NW);
label("$5$", (B + O)/2, NE);
label("$8$", (B + D)/2, S);
label("$y$", (A + C)/2, N);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, E);
label("$D$", D, W);
[/asy]

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