
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 that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the sequence $\left\{{\frac{{{S_n}}}{n}}\right\}$ forms an arithmetic sequence with a first term of $\frac{1}{3}$ and a common difference of $\frac{1}{3}$. If $\left[x\right]$ represents the greatest integer not exceeding $x$, such as $\left[0,1\right]=0$, $\left[1,9\right]=1$, then the sum of the first $35$ terms of the sequence $\{[a_{n}]\}$ is ____.

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