
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
Find all natural numbers \( n \geq 2 \) for which the equality \( 4 x_{n} + 2 y_{n} = 20 n^{2} + 13 n - 33 \) holds, where
\[ x_{n} = 1 \cdot 2 + 2 \cdot 3 + \cdots + (n-1) \cdot n \]
and
\[ y_{n} = 1^{2} + 2^{2} + 3^{2} + \cdots + (n-1)^{2}. \]

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