
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
Consider a sequence \( x_{n} \) such that \( x_{1} = x_{2} = 1 \), and \( x_{3} = \frac{2}{3} \). Suppose that \( x_{n} = \frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2} - x_{n-1} x_{n-3}} \) for all \( n \geq 4 \). Find the least \( n \) such that \( x_{n} \leq \frac{1}{10^{6}} \).

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