
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
Let the function $f(x)= \sqrt{3}\cos^2\omega x+\sin \omega x\cos \omega x+a$ where $\omega > 0$, $a\in\mathbb{R}$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $\dfrac{\pi}{6}$.

(Ⅰ) Find the smallest positive period of $f(x)$; (Ⅱ) If the minimum value of $f(x)$ in the interval $\left[-\dfrac{\pi}{3}, \dfrac{5\pi}{6}\right]$ is $\sqrt{3}$, find the value of $a$.

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