
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 vectors $\overrightarrow{a}=(A\sin \omega x,A\cos \omega x)$, $\overrightarrow{b}=(\cos \theta,\sin \theta)$, $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}+1$, where $A > 0$, $\omega > 0$, and $\theta$ is an acute angle. The distance between two adjacent symmetry centers of the graph of $f(x)$ is $\dfrac {\pi}{2}$, and when $x= \dfrac {\pi}{12}$, $f(x)$ reaches its maximum value of $3$.  
(Ⅰ) Find the analytical expression of $f(x)$;  
(Ⅱ) The graph of $f(x)$ is first translated downward by 1 unit to obtain the graph of a function, and then translated to the left by $\phi (\phi > 0)$ units to obtain the graph of $g(x)$. If $g(x)$ is an odd function, find the minimum value of $\phi$.

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