
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
If an integer can be expressed in the form of $a^{2}+b^{2}$ (where $a$ and $b$ are positive integers), then this number is called a "Fengli number." For example, $2$ is a "Fengli number" because $2=1^{2}+1^{2}$, and for example, $M=x^{2}+2xy+2y^{2}=\left(x+y\right)^{2}+y^{2}$ (where $x$ and $y$ are positive integers), so $M$ is also a "Fengli number." <br/> $(1)11$ is ______ a "Fengli number" (Yes or No); <br/> $(2)$ If $p=4x^{2}+mxy+2y^{2}-10y+25$ (where $x \gt y \gt 0$) is a "Fengli number," then $m=\_\_\_\_\_\_$.

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