
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 \( R_{n}=\frac{1}{2}\left(a^{n}+b^{n}\right) \), where \( a = 3 + 2\sqrt{2} \), \( b = 3 - 2\sqrt{2} \), and \( n=1, 2, \ldots \). What is the units digit of \( R_{12345} \)?

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