
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
The sum of the first  $n$  terms of the sequence   $$ 1,1+2,1+2+2^2,\ldots,1+2+\cdots+2^{k-1},\ldots $$   is of the form  $2^{n+R}+Sn^2+Tn+U$  for all  $n>0.$  Find  $R,S,T,$  and  $U.$ 

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