
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
Convex pentagon  $ABCDE$  is inscribed in circle  $\gamma$ . Suppose that  $AB=14$ ,  $BE=10$ ,  $BC=CD=DE$ , and  $[ABCDE]=3[ACD]$ . Then there are two possible values for the radius of  $\gamma$ . The sum of these two values is  $\sqrt{n}$  for some positive integer  $n$ . Compute  $n$ .

*Proposed by Luke Robitaille*

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