
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
Let us define a sequence with $S_n=\frac{1}{2}(p^n+q^n)$ where $p = 5 + 4\sqrt{2}$, $q = 5 - 4\sqrt{2}$, and $n=0,1,2,\dots$. Determine the units digit of $S_{6789}$.
A) 1
B) 3
C) 5
D) 7
E) 9

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