
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
Given the expression of the function $y=f\left(x\right)$ as $f(x)=\frac{1}{3^x+1}-a(a∈R)$.  
$(1)$ Determine the monotonicity of the function $y=f\left(x\right)$ on its domain and explain the reason;  
$(2)$ Does there exist a real number $a$ such that the function $y=f\left(x\right)$ is an odd function? Explain the reason.

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