
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 any set  $A = \{x_1, x_2, x_3, x_4, x_5\}$  of five distinct positive integers denote by  $S_A$  the sum of its elements, and denote by  $T_A$  the number of triples  $(i, j, k)$  with  $1 \le i < j < k \le 5$  for which  $x_i + x_j + x_k$  divides  $S_A$ .
Find the largest possible value of  $T_A$ .

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