
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) Given the curve $y=x+\ln x$, the tangent line at point $(1,1)$ intersects with the curve $y=ax^2+(a+2)x+1$. Find the value of $a$.

(2) Observe the following expressions: $1+\frac{1}{2^2}<\frac{3}{2}$, $1+\frac{1}{2^2}+\frac{1}{3^2}<\frac{5}{3}$, $1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}<\frac{7}{4}$. What can be inducted from these expressions?

(3) Given the parabola $C:y^2=4x$ with focus $F$, line $l_1:y=x-1$ intersects the parabola at points $A$ and $B$. From points $A$ and $B$, draw vertical lines to line $l_2:x=-2$, intersecting at points $D$ and $C$, respectively. Find the perimeter of quadrilateral $ABCD$.

(4) Given the function $f(x)=\frac{\ln x+(x-b)^2}{x}$ $(b\in\mathbb{R})$, there exists an $x\in[\frac{1}{2},2]$ such that $f(x)+xf'(x)>0$. Find the range of values for the real number $b$.

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