
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 vectors $a=(1+\cos \omega x,1)$ and $b=(1,a+\sqrt{3}\sin \omega x)$ (where $\omega$ is a constant and $\omega > 0$), the function $f(x)=a\cdot b$ has a maximum value of $2$ on $\mathbb{R}$.

(Ⅰ) Find the value of the real number $a$;

(Ⅱ) If the graph of the function $y=f(x)$ is translated to the right by $\dfrac{\pi}{6\omega}$ units to obtain the graph of the function $y=g(x)$, and if $y=g(x)$ is an increasing function on the interval $\left[0,\dfrac{\pi}{4}\right]$, find the maximum value of $\omega$.

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