
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
Find the minimum value of the expression
\[
\frac{\sin^8 x + \cos^8 x + 1}{\sin^6 x + \cos^6 x + 1}
\]
over all real values \( x \).

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