
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
We agree that if the graph of a linear function intersects the graph of a quadratic function at two points, and both points are on the coordinate axis, then the quadratic function and the linear function are considered "closely related" functions, and these two points are called the "related points" of the two functions. According to this agreement, answer the following questions:
$(1)$ Determine whether the functions $y=2x+1$ and $y=2x^{2}+3x+1$ are "closely related" functions. If they are, find their "related points";
$(2)$ The graph of the function $y=x+3c(c$ is a constant and $c \gt 0)$ intersects the $x$-axis at point $A$, and its "closely related" function $y=ax^{2}+2bx+3c$ intersects the $x$-axis at another point $B$ such that $OA=3OB$. Find the value of $b$;
$(3)$ If the function $y_{1}=x+1$ has a "closely related" function $y_{2}=mx^{2}+nx+t$, and the function $y=y_{2}-y_{1}-m$ has a minimum value of $m$ for $m\leqslant x\leqslant m+1$, find the analytical expression of the function $y$.

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