
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 $\overrightarrow{a}=({\sin x, -2\cos x})$, $\overrightarrow{b}=({2\sqrt{3}\cos x, \cos x})$, and a function $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}+1$.  
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.  
$(2)$ Shorten the abscissa of all points on the graph of $f(x)$ to half of the original length, keep the ordinate unchanged, and then shift left by $\frac{\pi}{12}$ units to obtain the graph of a function $g(x)$. Find the range of $g(x)$ on $[-\frac{\pi}{12}, \frac{\pi}{8}]$.

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