
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 the function $f(x)=\sin kx \sin^k x + \cos kx \cos^k x - \cos^k 2x$, where $k$ is a constant and $x \in \mathbb{R}$.

(1) When $k=1$, find the interval where the function $f(x)$ is monotonically increasing.

(2) When $k=1$, find the maximum value of the function $g(x)=\frac{f(x)}{a+f^2(x)}$ on the interval $(0, \frac{\pi}{3}]$, where $a > 0$ is a constant.

(3) Determine if there exists a $k \in \mathbb{N}^*$ such that the function $f(x)$ is a constant function. If it exists, find the value of $k$ and provide a proof; if not, explain the reason.

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