
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
(1) The negation of the proposition "There exists an $x \in \mathbf{R}$, such that $x^2 - ax + 1 < 0$" is _______.

(2) Given $\vec{a} =(1,3)$, $\vec{b} =(-2,k)$, and $(\vec{a} +2\vec{b} )\parallel(3\vec{a} -\vec{b} )$, find the real number $k=$ _______.

(3) Given $\sin 2\alpha-2=2\cos 2\alpha$, find $\sin^2\alpha+\sin 2\alpha=$ _______.

(4) Given that the line $y=b$ intersects the functions $f(x)=2x+3$ and $g(x)=ax+\ln x$ at points $A$ and $B$ respectively, if the minimum value of $|AB|$ is $2$, find $a+b=$ _______.

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