
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
【Reading comprehension question】Read the following material and complete the corresponding tasks:

Symmetrical Expression
In an expression containing multiple letters, if the expression remains unchanged when any two letters swap positions, the expression is called a symmetrical expression.
For example: In the expression $abc$, swapping any two letters gives the expressions $bac$, $acb$, $cba$. Since $abc = bac = acb = cba$, $abc$ is a symmetrical expression.
However, swapping the letters $a$ and $b$ in the expression $a-b$ gives the expression $b-a$. Since $a-b \neq b-a$, $a-b$ is not a symmetrical expression.

Tasks:
$(1)$ Among the following expressions, the symmetrical expressions are ______ $(Fill in the serial number)$:
① $a+b+c$
② $a^{2}+b^{2}$
③ $a^{2}b$
④ $\frac{a}{b}$
$(2)$ Write a monomial containing only the letters $x$ and $y$ such that the monomial is a symmetrical expression and has a degree of $6$;
$(3)$ Given $A=2a^{2}+4b^{2}$, $B=a^{2}-2ab$, find $A+2B$, and directly determine if the result is a symmetrical expression.

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