
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
A sequence satisfies \( a_{0}=\frac{1}{4} \), and for natural numbers \( n \), \( a_{n+1}=a_{n}^{2}+a_{n} \). Find the integer part of \( \sum_{n=0}^{2011} \frac{1}{a_{n}+1} \).

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