
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 the sequence defined by:
$$
u_{0}=1, \quad u_{1}=2, \quad u_{2}=0 \quad \text{and}
$$
$$
u_{n}=2 u_{n-1}+u_{n-2}-2 u_{n-3} \quad(n \geqslant 3),
$$
find the general term formula for the sequence $\left\{u_{n}\right\}_{n \geqslant 0}$.

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