
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
Given a sequence \(\left\{x_{n}\right\}\) satisfying \(x_{1} = \frac{1}{3}\) and \(x_{k+1} = x_{k} + x_{k}^{2}\) for \(k = 1, 2, \cdots\), determine between which two consecutive integers the value of \(\frac{1}{x_{1} + 1} + \frac{1}{x_{2} + 1} + \cdots + \frac{1}{x_{40} + 1}\) lies.

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