
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
The definition of an even function: Generally, if for any real number $x$ in the domain $D$ of the function $f(x)$, we have $f(-x) = f(x)$, then the function $f(x)$ is called an even function. The definition of an odd function: Generally, if for any real number $x$ in the domain $D$ of the function $f(x)$, we have $f(-x) = -f(x)$, then the function $f(x)$ is called an odd function.  
$(1)$ Why is the domain of a function with parity symmetry always symmetric about the origin?  
$(2)$ What is the difference between determining whether a function has parity symmetry and determining that a function does not have parity symmetry?  
$(3)$ Why is it necessary for an odd function to have $f(0) = 0$ when it is defined at the origin?

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