
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
In the sequence ${a_n}$, the first term is ${a_1=\frac{1}{3}}$, and the sum of the first $n$ terms, denoted as ${S_n}$, satisfies ${S_{n+1}=S_n+\frac{2^n}{3}\left( n\in N^* \right)}$.

(I) Find the general term formula for the sequence ${a_n}$;

(II) Let ${b_n=\log\left(S_n+\frac{1}{3}\right)}$, find the sum of the first $n$ terms for the sequence ${b_n}$, denoted as ${T_n}$.

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