
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  $ABC$  be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let  $N = 2017$ , and define the  $A$ -*ntipodes* to be the points  $A_1,\dots, A_N$  to be the points on segment  $BC$  such that  $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$ , and similarly define the  $B$ ,  $C$ -ntipodes. A line  $\ell_A$  through  $A$  is called a *qevian* if it passes through an  $A$ -ntipode, and similarly we define qevians through  $B$  and  $C$ . Compute the number of ordered triples  $(\ell_A, \ell_B, \ell_C)$  of concurrent qevians through  $A$ ,  $B$ ,  $C$ , respectively.

*Proposed by Brandon Wang*

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