
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 $\overrightarrow{a}=(3 \sqrt {3}\sin x, \sqrt {3}\cos x)$ and $\overrightarrow{b}=(\cos x, \sqrt {3}\cos x)$, and define $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.  
(1) Find the interval of monotonic decrease for $f(x)$.  
(2) When $x\in\left[- \dfrac {\pi}{3}, \dfrac {\pi}{3}\right]$, the maximum value of $g(x)=f(x)+m$ is $\dfrac {11}{2}$. Find the minimum value of $g(x)$ and the corresponding value of $x$.

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