
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 vectors $\overset{⇀}{a}=(1,x)$, $\overset{⇀}{b}=(x-1,2)$, if $\overset{⇀}{a}\parallel \overset{⇀}{b}$, then $x=$ ______.

(2) For a function $f(x)$ defined on $\mathbb{R}$ with derivative $f'(x)$, if $f(x)=x^3-f'(1)x^2+1$, then $f'(1)=$ ______.

(3) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=\sqrt{2}c$ and $\sin A+\sqrt{2}\sin C=2\sin B$, then $\cos A=$ ______.

(4) Given the function $y=x+\ln x$ has a tangent line $l$ at point $(1,1)$. If $l$ is tangent to the curve $y=ax^2+(a+2)x+1$, then $a=$ ______.

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