
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
Let the function $y=f(x)$ have the domain of $\mathbb{R}$, and satisfy $f(x+y)=f(x)+f(y)$, $f\left(\frac{1}{3}\right)=1$, and when $x > 0$, $f(x) > 0$.

$(1)$ Find the value of $f(0)$; $(2)$ Determine the parity of the function; $(3)$ If $f(x)+f(2+x) < 2$, find the range of values for $x$.

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