
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 $f(x)=\cos \omega x (\omega > 0)$, the distance between two adjacent axes of symmetry on its graph is $\dfrac{\pi}{2}$,

(Ⅰ) Find the intervals of monotonicity for $f(x+ \dfrac{\pi}{6})$ in the interval $\left[- \dfrac{\pi}{6}, \dfrac{2\pi}{3}\right]$;

(Ⅱ) If $\alpha \in \left( \dfrac{5\pi}{12}, \dfrac{\pi}{2} \right)$ and $f\left(\alpha+ \dfrac{\pi}{3}\right)= \dfrac{1}{3}$, find the value of $\sin 2\alpha$.

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