
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 $\overrightarrow{m}=(a+1,\sin x), \overrightarrow{n}=(1,4\cos (x+ \frac {π}{6}))$, define the function $g(x)= \overrightarrow{m} \cdot \overrightarrow{n}$ where $a \in \mathbb{R}$ is a constant.
(1) Find the smallest positive period of $g(x)$ for any real number $x$;
(2) If the sum of the maximum and minimum values of $g(x)$ on $[0, \frac {π}{3})$ is $7$, find the value of $a$.

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