
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 $f(x) = x^2 - ax + \ln x$, where $a \in \mathbb{R}$.
1. When $a=3$, find the minimum value of the function $f(x)$.
2. Let $g(x) = x^2 - f(x)$, determine if there exists a real number $a$ such that when $x \in [1, e]$ (where $e$ is the base of the natural logarithm), the function $g(x)$ attains its minimum value of 1. If such an $a$ exists, find its value; if not, explain why.

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