
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
On the coordinate plane is given the square with vertices  $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$ . For every  $n\in\mathbb N$ , point  $T_{n+4}$  is defined as the midpoint of the segment  $T_nT_{n+1}$ . Determine the coordinates of the limit point of  $T_n$  as  $n\to\infty$ , if it exists.

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