
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
Given vector  $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$  and recursively defined sequence of vectors  $\{\mathbf{v}_n\}_{n\geq 0}$   $$ \mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1} $$  Evaluate the value of infinite series  $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$ .

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