
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
Consider the sequence given by $R_n=\tfrac{1}{2}(a^n+b^n)$ where $a=3+\sqrt{5}$ and $b=3-\sqrt{5}$, and $n=0,1,2,\cdots$. Determine the units digit of $R_{10001}$.
A) 0
B) 1
C) 3
D) 7
E) 9

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