
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
In the Cartesian coordinate system, the "Manhattan distance" between two points $P(x_{1}, y_{1})$ and $Q(x_{2}, y_{2})$ is defined as $|x_{1}-x_{2}|+|y_{1}-y_{2}|$, denoted as $|PQ|$. For example, the "Manhattan distance" between points $P(-1,-2)$ and $Q(2,4)$ is $9$, denoted as $|PQ|=|(-1)-2|+|(-2)-4|=9$.
$(1)$ Given that the moving point $P$ is on the line $y=1-x$, and point $O(0,0)$, if $|PO|\leqslant 1$, find the range of the abscissa $x$ of point $P$;
$(2)$ Given that the moving point $P$ is on the line $y=2x-2$, and the moving point $Q$ is on the graph of the function $y=x^{2}$, find the minimum value of $|PQ|$;
$(3)$ Given that the moving point $Q$ is on the graph of the function $y=x^{2}$ where $x\in [-2,2]$, and point $P(a,b)$, the maximum value of $|PQ|$ is denoted as $M(a,b)$. For example, when the coordinates of point $P$ are $(0,0)$, $M(0,0)=6$. Find the minimum value of $M(a,b)$ and determine the coordinates of point $P$ at that time.

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