
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 $\overrightarrow{a}=( m+1,\sin x )$, $\overrightarrow{b}=( 1,4\cos \left( x+\frac{\pi }{6} \right) )$, and let the function $g\left( x \right)=\overrightarrow{a}\cdot \overrightarrow{b}$, where $m\in \mathbb{R}$ and $m$ is a constant.

$(1)$ If $x$ is any real number, find the smallest positive period of $g\left( x \right)$;

$(2)$ If the sum of the maximum and minimum values of $g\left( x \right)$ in the interval $\left[ 0,\frac{\pi }{3} \right)$ is $7$, find the value of $m$.

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