
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
In the Cartesian coordinate system, given that $\odot O$ is a circle with center at the origin $O$ and radius $2$, point $A(2,0)$, and angle $x$ (in radians) with initial side as the ray $OA$ and terminal side intersecting circle $\odot O$ at point $B$. The ordinate $y$ of point $B$ is a function of angle $x$, denoted as $y=f(x)$.  
$(1)$ Write down the analytical expression of the function $y=f(x)$.  
$(2)$ Multiply the abscissa of each point on the graph of the function $y=f(x)$ by $2$ (keeping the ordinate unchanged), then shift the resulting graph to the left by $\frac{{2\pi}}{3}$ units to obtain the graph of the function $y=g(x)$. Find the maximum and minimum values of the function $y=g(x)$ on the interval $[0,2\pi]$, and write down the values of the independent variable $x$ at which these extreme values occur.

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