
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 $f(x)=\cos \left(x+\dfrac{\pi }{3}\right) \cdot \cos x-\dfrac{1}{4}$.

(I) Find the value of $f\left(\dfrac{\pi }{3}\right)$ and the interval where $f(x)$ is monotonically increasing.

(II) If the function $f(x+\theta)$ is an odd function for $\theta \in \left[0,\dfrac{\pi }{2}\right]$, find the range of the function $y=\left[f\left(x+\theta \right)\right]^{2}$.

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