
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 have learned various applications of the multiplication formula $\left(a\pm b\right)^{2}=a^{2}\pm 2ab+b^{2}$, and we can use this knowledge to answer the following question: Find the minimum value of the algebraic expression $x^{2}+4x+5$. The solution is as follows:

Solution: $x^{2}+4x+5=x^{2}+4x+4+1=\left(x+2\right)^{2}+1$,  
$(x+2)^{2}\geqslant 0$, therefore when $x=-2$, the value of $\left(x+2\right)^{2}$ is minimum, which is $0$,  
Hence, $\left(x+2\right)^{2}+1\geqslant 1$, therefore when $\left(x+2\right)^{2}=0$, the value of $\left(x+2\right)^{2}+1$ is minimum, which is $1$,  
Therefore, the minimum value of $x^{2}+4x+5$ is $1$.  
Please answer the following questions based on the above method.  
$(1)$ Knowledge Review: When $x=$______ , the minimum value of the algebraic expression $x^{2}-4x+15$ is ______;  
$(2)$ Application of Knowledge: If $y=-x^{2}+6x-15$, when $x=$______ , $y$ has the most ______ value (fill in "large" or "small"), and this value is ______;  
$(3)$ Knowledge Extension: If $-x^{2}+5x+y+10=0$, find the minimum value of $y+x$.

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