
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 sequence $\{a\_n\}$ has a sum of the first $n$ terms denoted as $S\_n$, with $a\_1 = \frac{2}{3}$ and $a\_{n+1} - S\_n = \frac{2}{3}$. Using $[x]$ to represent the largest integer not greater than $x$ (e.g., $[-0.4] = -1$ and $[1.6] = 1$), let $b\_n = [a\_n]$. Determine the sum of the first $2n$ terms of the sequence $\{b\_n\}$.

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