
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 functions $f(x)=x^{2}-2ax+1$ and $g(x)=x-a$, where $a > 0$.

$(1)$ For any $x \in \mathbb{R}$, $f(x) > g(x)$ always holds, find the range of the real number $a$;

$(2)$ For any $x_{1} \in [-2,-1]$, $x_{2} \in [2,4]$, $f(x_{1}) > g(x_{2})$ always holds, find the range of the real number $a$;

$(3)$ There exist $x_{1} \in [-2,-1]$, $x_{2} \in [2,4]$, such that $f(x_{1}) > g(x_{2})$ holds, find the range of the real number $a$.

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