
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 \(\left\{x_{n}\right\}\) is defined as follows: \(x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}\). Calculate 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}
user