
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 the function $y=f(x), x \in D$;

$(1)$ When the function $f(x)=x^{2}-2ax+5$ $(a > 1)$ and both the domain and range of $f(x)$ are $[1,a]$, find the value of the real number $a$;

$(2)$ When the function $f(x)=\arccos \left(x-\frac{1}{2}\right)$, $D=\left[ \frac{1}{3}, \frac{2}{3} \right]$ exists $x \in D$, such that the inequality $\sqrt{f(x) \cdot f(1-x)} \geqslant a$ holds, find the range of the real number $a$;

$(3)$ When the function $f(x)=\frac{a}{x}-x$, $D=(0, 1)$, for any $x \in D$, the inequality $f(x) \cdot f(1-x) \geqslant 1$ always holds, find the range of the real number $a$.

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