
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 $f(x)=\left\{\begin{array}{l}{\frac{1}{a}x, 0≤x≤a}\\{\frac{1}{1-a}(1-x), a＜x≤1}\end{array}\right.$, where $a$ is a constant and $a\in \left(0,1\right)$. A new definition: if $x_{0}$ satisfies $f(f(x_{0}))=x_{0}$, but $f(x_{0})\neq x_{0}$, then $x_{0}$ is called a turning point of $f(x)$.  
$(1)$ When $a=\frac{1}{2}$, find the value of $f(f(\frac{4}{5}))$ and determine if $\frac{4}{5}$ is a turning point;  
$(2)$ When $x\in \left(a,1\right]$, find the analytical expression of the function $y=f(f(x))$, and find the turning points of $f(x)$.

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