
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
Define the sequence  $(x_{n})$ :  $x_{1}=\frac{1}{3}$  and  $x_{n+1}=x_{n}^{2}+x_{n}$ . Find  $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$ , wehere  $[$   $]$  denotes the integer part.

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