
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 $f(x) = x^2 - 2ax - 1 + a$, where $a \in \mathbb{R}$.

$(1)$ If $a=2$, find the minimum value of the function $y = \frac{f(x)}{x}$ $(x > 0)$.

$(2)$ For any $x \in [0,2]$, the inequality $f(x) \leqslant a$ holds, find the range of the real number $a$.

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