
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
Given the expression $M=\left(a+4\right)x^{3}+6x^{2}-2x+5$ is a quadratic polynomial in terms of $x$, and the coefficient of the quadratic term is $b$. Points $A$ and $B$ on the number line correspond to the numbers $a$ and $b$, respectively.<br/>$(1)$ $a=$______,$b=$______;<br/>$(2)$ If point $P$ starts from point $A$ and moves in the positive direction of the number line at a speed of $2$ units per second, after $t$ seconds, the distance from point $P$ to point $B$ is half the distance from point $P$ to point $A$. Find the value of $t$ at this moment;<br/>$(3)$ If point $P$ starts from point $A$ and moves in the negative direction of the number line at a speed of $\frac{1}{2}$ unit per second, and two seconds later, point $Q$ starts from point $B$ and moves in the negative direction of the number line at a speed of $2$ units per second. When the distance between points $P$ and $Q$ is $8$, find the time of point $P$'s movement.

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