
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 an integer  $k$  let  $T_k$  denote the number of  $k$ -tuples of integers  $(x_1,x_2,...x_k)$  with  $0\le x_i < 73$  for each  $i$ , such that  $73|x_1^2+x_2^2+...+x_k^2-1$ . Compute the remainder when  $T_1+T_2+...+T_{2017}$  is divided by  $2017$ .

[i]Proposed by Vincent Huang

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