
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
Let the real-valued sequence $\left\{x_{n}\right\}$ satisfy the following conditions: $x_{0}=0$, $x_{2}=\sqrt[3]{2}x_{1}$, $x_{3}$ is a positive integer, and $x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for $n \geq 2$. How many integer terms are there in such sequences at the minimum?

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