
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
New definition: Given that $y$ is a function of $x$, if there exists a point $P(a, a+2)$ on the graph of the function, then point $P$ is called a "real point" on the graph of the function. For example, the "real point" on the line $y=2x+1$ is $P(1,3)$.

$(1)$ Determine whether there is a "real point" on the line $y=\frac{1}{3}x+4$. If yes, write down its coordinates directly; if not, explain the reason.

$(2)$ If there are two "real points" on the parabola $y=x^{2}+3x+2-k$, and the distance between the two "real points" is $2\sqrt{2}$, find the value of $k$.

$(3)$ If there exists a unique "real point" on the graph of the quadratic function $y=\frac{1}{8}x^{2}+\left(m-t+1\right)x+2n+2t-2$, and when $-2\leqslant m\leqslant 3$, the minimum value of $n$ is $t+4$, find the value of $t$.

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