
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
Let $y=f\left(x\right)$ be a function defined on $R$. If there exists an interval $\left[a,b\right]$ and $x_{0}\in \left(a,b\right)$ such that $y=f\left(x\right)$ strictly decreases on $[a,x_{0}]$ and strictly increases on $[x_{0},b]$, then $y=f\left(x\right)$ is called a "valley function" and $\left[a,b\right]$ is called a "valley interval".  
$(1)$ Determine which of the following functions are valley functions. If yes, indicate the valley point; if not, provide the reason.  
① $y=2|x|$, ② $y=x+\cos x$;  
$(2)$ Given a real number $m \gt 0$, $y=x^{2}-2x-m\ln \left(x-1\right)$ is a valley function, and $\left[2,4\right]$ is a valley interval of it. Find the range of $m$;  
$(3)$ Let $p$, $q\in R$, $h\left(x\right)=-x^{4}+px^{3}+qx^{2}+\left(4-3p-2q\right)x$. Suppose the function $y=h\left(x\right)$ is a valley function, and $\left[a,b\right]$ is a valley interval of it, with the maximum value of $b-a$ denoted as $L\left(p,q\right)$. If $h\left(1\right)\leqslant h\left(2\right)$ and $h\left(1\right)\leqslant 0$, find the minimum value of $L\left(p,q\right)$.

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