
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
Let  $ABCD$  be a cyclic quadrilateral with  $AB = 5$ ,  $BC = 10$ ,  $CD = 11$ , and  $DA = 14$ . The value of  $AC + BD$  can be written as  $\tfrac{n}{\sqrt{pq}}$ , where  $n$  is a positive integer and  $p$  and  $q$  are distinct primes. Find  $n + p + q$ .

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