
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
For the given quadrilateral, calculate the number of possible integer values for the length of the diagonal, represented by the dashed line.

[asy]
draw((0,0)--(4,4)--(10,-1)--(6,-7)--cycle,linewidth(0.7));
draw((0,0)--(10,-1),dashed);
label("6",(2,2),NW);
label("12",(7,1.5),NE);
label("14",(8, -4),SE);
label("10",(3,-3.5),SW);
[/asy]

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