
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
Read the following materials:<br/>【Material 1】If $m^{2}-2mn+2n^{2}-8n+16=0$, find the values of $m$ and $n$.<br/>Solution: $\because m^{2}-2mn+2n^{2}-8n+16=0$,<br/>$\therefore (m^{2}-2mn+n^{2})+(n^{2}-8n+16)=0$,<br/>$\therefore \left(m-n\right)^{2}+\left(n-4\right)^{2}=0$,<br/>$\therefore \left(m-n\right)^{2}=0$,$\left(n-4\right)^{2}=0$,<br/>$\therefore n=4$,$m=4$.<br/>【Material 2】The conclusion "$a\geqslant 0$" is very useful in mathematics, sometimes we need to complete the algebraic expression in a perfect square form. For example: $m^{2}+8m+17=m^{2}+8m+16+1=\left(m+4\right)^{2}+1$.<br/>$\because \left(m+4\right)^{2}\geqslant 0$,<br/>$\therefore \left(m+4\right)^{2}+1\geqslant 1$,<br/>$\therefore m^{2}+8m+17\geqslant 1$. Hence $m^{2}+8m+17$ has a minimum value of $1$. Read the materials and explore the following questions:<br/>$(1)$ Given $x^{2}-2xy+2y^{2}+6y+9=0$, find the value of $xy$;<br/>$(2)$ Regardless of the value of $m$, the algebraic expression $m^{2}+6m+13$ always has a minimum value, find the minimum value.

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