
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 \(\{x_{n}\}\) is defined as follows: \(x_{1} = \frac{1}{2}\), \(x_{k+1} = x_{k}^{2} + x_{k}\). Find the integer part of the sum \(\frac{1}{x_{1}+1} + \frac{1}{x_{2}+1} + \cdots + \frac{1}{x_{100}+1}\).

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