
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 $f(x)=ax^2 - (2a-1)x - \ln x$ (where $a$ is a constant, $a\neq 0$),
1. When $a < 0$, find the maximum value of the function $f(x)$ in the interval $[1,2]$;
2. Denote the graph of function $f(x)$ as curve $C$, and let $A(x_1, y_1)$ and $B(x_2, y_2)$ be two distinct points on curve $C$. If $M$ is the midpoint of segment $AB$, and a perpendicular line to the $x$-axis is drawn through point $M$ that intersects curve $C$ at point $N$, determine whether the tangent to curve $C$ at point $N$ is parallel to line $AB$ and explain the reasoning.

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