
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  $A_1B_1C_1$  be a triangle with  $A_1B_1 = 16, B_1C_1 = 14,$  and  $C_1A_1 = 10$ . Given a positive integer  $i$  and a triangle  $A_iB_iC_i$  with circumcenter  $O_i$ , define triangle  $A_{i+1}B_{i+1}C_{i+1}$  in the following way:

(a)  $A_{i+1}$  is on side  $B_iC_i$  such that  $C_iA_{i+1}=2B_iA_{i+1}$ .
(b)  $B_{i+1}\neq C_i$  is the intersection of line  $A_iC_i$  with the circumcircle of  $O_iA_{i+1}C_i$ .
(c)  $C_{i+1}\neq B_i$  is the intersection of line  $A_iB_i$  with the circumcircle of  $O_iA_{i+1}B_i$ .

Find \[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \]

Note:  $[K]$  denotes the area of  $K$ .

*Proposed by Yang Liu*

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