
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
When studying knowledge about polynomials, Lily found an interesting phenomenon: for the polynomial $x^{2}-4x+7$, since $x^{2}-4x+7=\left(x-2\right)^{2}+3$, when $x-2$ takes any pair of numbers that are opposite to each other, the value of the polynomial $x^{2}-4x+7$ is equal. For example, when $x-2=\pm 1$, i.e., $x=3$ or $1$, the value of $x^{2}-4x+7$ is both $4$; when $x-2=\pm 2$, i.e., $x=4$ or $0$, the value of $x^{2}-4x+7$ is both $7$. Therefore, Lily gives a definition: for a polynomial in terms of $x$, if the value of the polynomial is equal when $x-m$ takes any pair of numbers that are opposite to each other, then the polynomial is said to be symmetric about $x=m$. For example, $x^{2}-4x+7$ is symmetric about $x=2$. Please solve the following problems based on Lily's thought process and using this definition:
$(1)$ The polynomial $x^{2}-2x+5$ is symmetric about $x=$______.
$(2)$ If the polynomial $x^{2}+2nx+3$ is symmetric about $x=5$, find the value of $n$.
$(3)$ If the polynomial $(x^{2}+6x+9)(x^{2}-4x+4)$ is symmetric about $x=a$, find the real number $a$.

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