
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
We call the *tribi* of a positive integer  $k$  (denoted  $T(k)$ ) the number of all pairs  $11$   in the binary representation of  $k$ . e.g   $$ T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2. $$  Calculate  $S_n=\sum_{k=1}^{2^n}T(K)$ .

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