
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 $f(x)$ be a function defined on $\mathbb{R}$ with the smallest positive period of $\frac{7\pi}{6}$, and on $\left[-\frac{5\pi}{6}, \frac{\pi}{3}\right]$ it is defined as $f\left(x\right)=\begin{cases}\sin x,x\in\left[-\frac{5\pi}{6},0\right) & \\\cos x+a,x\in\left[0,\frac{\pi}{3}\right] & \end{cases}$. Determine the value of $a$ and $f\left(-\frac{16\pi}{3}\right)$.

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