
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 $\{a_n\}$ satisfies $a_1= \frac {4}{3}, a_{n+1}-1=a_{n}(a_{n}-1), n\in\mathbb{N}^*$ and $S_{n}= \frac {1}{a_{1}}+ \frac {1}{a_{2}}+\ldots+ \frac {1}{a_{n}}$. The set of all possible integer parts of $S_{n}$ is __________.

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