
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 \( k \) be an integer greater than 1, and let \( a \) be a root of the equation \( x^{2} - kx + 1 = 0 \). For any natural number \( n \) greater than 10, the units digit of \( a^{2^{n}} + a^{-2^{n}} \) is always 7. What is the units digit of \( k \)?

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