
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 $\left\{a_{n}\right\}$ that satisfies:
$$
a_{1}=a_{2}=a_{3}=1, a_{4}=-1, a_{5}=0,
$$

and for any $n \in \mathbf{Z}_{+}$, it holds that
$$
a_{n+5}=3 a_{n+4}-4 a_{n+3}+4 a_{n+2}-3 a_{n+1}+a_{n}.
$$

Find the general formula for the sequence $\left\{a_{n}\right\}$.

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