
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 parabola $y=-2x^{2}+bx+c$, it is tangent to the line $y=x-3$ at point $(2,-1)$. Find the value of $b+c$.

(2) Let $M$ and $N$ be two points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$. The product of the slopes of lines $OM$ and $ON$ is $-\frac{b^{2}}{a^{2}}$ ($O$ is the origin). For any point $P$ in the plane, the following results are obtained:

- If $\overrightarrow{OP}=\overrightarrow{OM}+\overrightarrow{ON}$, the trajectory equation of point $P$ is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=2$.
- If $\overrightarrow{OP}=2\overrightarrow{OM}+\overrightarrow{ON}$, the trajectory equation of point $P$ is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=5$.
- If $\overrightarrow{OP}=\overrightarrow{OM}+2\overrightarrow{ON}$, the trajectory equation of point $P$ is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=5$.
- If $\overrightarrow{OP}=3\overrightarrow{OM}+\overrightarrow{ON}$, the trajectory equation of point $P$ is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=10$.
- If $\overrightarrow{OP}=\overrightarrow{OM}+3\overrightarrow{ON}$, the trajectory equation of point $P$ is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=10$.

Based on these results, find the trajectory equation of point $P$ when $\overrightarrow{OP}=m\overrightarrow{OM}+n\overrightarrow{ON} (m,n\in\mathbb{N}^{\ast})$.

(3) Let the parabola be $y^{2}=2px (p > 0)$, with focus $F$ and directrix $l$. A line passing through $F$ intersects the parabola at points $A$ and $B$, and the directrix at point $C$. Point $A$ is above the $x$-axis, $AK\perp l$, and the foot of the perpendicular is $K$. If $|BC|=2|BF|$ and $|AF|=4$, find the area of $\triangle AKF$.

(4) For a function $f(x)$ defined on $D$, if there exist two lines $y=kx+m_{1}$ and $y=kx+m_{2}$ with a distance of $d$ such that $kx+m_{1}\leqslant f(x)\leqslant kx+m_{2}$ holds for any $x\in D$, then $f(x)(x\in D)$ is said to have a width-$d$ channel. Given the following functions:

- $\quad\quad①f(x)=\frac{1}{x}$
- $\quad\quad②f(x)=\sin x$
- $\quad\quad③f(x)=\sqrt{x^{2}-1}$
- $\quad\quad④f(x)=\frac{\ln x}{x}$

Find the function(s) whose channel width can be $1$ in the interval $[1,+\infty)$.

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