
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 quadrilateral shown below, determine how many different positive integer lengths could be the length of the diagonal represented by the dashed line.

[asy]
draw((0,0)--(4,6)--(11,2)--(6,-7)--cycle,linewidth(0.7));
draw((0,0)--(11,2),dashed);
label("6",(2,3),NW);
label("9",(7.5,4),NE);
label("14",(8.5, -2.5),SE);
label("10",(3,-2.5),SW);
[/asy]

Label the quadrilateral vertices as $A(0,0), B(4,6), C(11,2), and D(6,-7)$. We seek the number of possible integer values for the diagonal $AC$.

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