
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
The sequences \(\{x_{n}\}\), \(\{y_{n}\}\) are defined by the initial conditions \(x_{1} = 11, y_{1} = 7\) and the recurrence relations \(x_{n+1} = 3x_{n} + 2y_{n}\), \(y_{n+1} = 4x_{n} + 3y_{n}\), for \(n \in \mathbb{N}\). Find the remainder when \(y_{1855}^{2018} - 2x_{1855}^{2018}\) is divided by 2018.

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