
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
Using the sequence defined by the recursion \( x_{n+1} = x_{n}^{2} - x_{n} + 1 \), construct the infinite series \(\sum_{i=1}^{\infty} \frac{1}{x_{i}}\). What is the sum of this series if \( a) x_{1} = \frac{1}{2} \) and \( b) x_{1} = 2 \)?

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