
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
Define: For linear functions $y_{1}=ax+b$ and $y_{2}=cx+d$, we call the function $y=m\left(ax+b\right)+n\left(cx+d\right)\left(ma+nc\neq 0\right)$ the "composite function" of functions $y_{1}$ and $y_{2}$. <br/> $(1)$ If $m=3$ and $n=1$, determine whether the function $y=5x+2$ is the "composite function" of functions $y_{1}=x+1$ and $y_{2}=2x-1$, and explain the reason; <br/> $(2)$ Let the graphs of the functions $y_{1}=x-p-2$ and $y_{2}=-x+3p$ intersect at point $P$. Find the coordinates of point $P$ (expressed in terms of $p$); <br/> $(3)$ Under the conditions of $(2)$, if $m+n \gt 1$, and point $P$ is above the graph of the "composite function" of functions $y_{1}$ and $y_{2$, find the range of values for $p$.

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