
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
Given a sequence of positive terms ${a_n}$ with the sum of the first $n$ terms denoted as $S_n$. It is known that $S_n$ is the arithmetic mean of $a_n^2$ and $a_n$.
(I) Find the general term formula for the sequence ${a_n}$.
(II) If ${a_{k_n}}$ is a member of ${a_1, a_2, …a_n, …}$ and forms a geometric sequence ${a_{k_1}, a_{k_2}, …, a_{k_n}, …}$, find the sum of the first $n$ terms, $T_n$, for the sequence ${k_n}$ when $k_1=2$ and $k_2=4$.

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