
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
The function f(x) is defined as follows:

$$
f(x) =
\begin{cases}
x^2 - 2ax + a & ,x \in [1, +\infty) \\
2x + \frac{a}{x} & ,x \in (0, 1)
\end{cases}
$$

1. Find the range of real number $a$ such that the function is increasing on the interval $(0, +\infty)$.
2. Find the range of real number $a$ such that the equation $f(x) = 1$ has three distinct real roots.
3. Determine if there exists a real number $a$ such that the function $f(x) \geq x - 2a$ always holds. If it exists, find the range of $a$; if not, explain the reason.

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