
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 the function \( f(x) = \sin^4 \left( \frac{kx}{10} \right) + \cos^4 \left( \frac{kx}{10} \right) \), where \( k \) is a positive integer. If for any real number \( a \), the set \(\{ f(x) \mid a < x < a+1 \} = \{ f(x) \mid x \in \mathbf{R} \}\), then find the minimum value of \( k \).

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