
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
From the relation \( x_{n-1}=\frac{x_{n}+x_{n-1}+x_{n-2}}{3} \), we obtain: \( 2 x_{n-1}=x_{n-2}+x_{n} \) for all \( n \geq 3 \). This means that the given sequence is an arithmetic progression. Let the common difference of the progression be \( d \). Then, \(\frac{x_{300}-x_{33}}{x_{333}-x_{3}}=\frac{(300-33)d}{330d}=\frac{89}{110}\).

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