
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 that $a_n$ is a ratio-difference sequence satisfying $\frac{a_{n+2}}{a_{n+1}} - \frac{a_{n+1}}{a_{n}}=d \ (n \in \mathbb{N^*}, d \text{ is a constant})$, and the initial values $a_1=a_2=1$ and $a_3=2$, find the last digit of $\frac{a_{2009}}{a_{2006}}$.

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