
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
Given that the range of the function $y=\sin^4 x + \cos^4 x$ $(x \in \mathbb{R})$ is $[\frac{1}{2}, 1]$, then  
(1) the range of the function $y=\sin^6 x + \cos^6 x$ $(x \in \mathbb{R})$ is ___________;  
(2) by analogy, the range of the function $y=\sin^{2n}x+\cos^{2n}x$ $(n \in \mathbb{N}^*)$ is ___________.

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