
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
Define the operation: $f(n)=\frac{1}{{2}^{n}+1}-\frac{1}{{2}^{n+1}+1}$, for example: $f(1)=\frac{1}{{2}^{1}+1}-\frac{1}{{2}^{2}+1}=\frac{1}{3}-\frac{1}{5}=\frac{2}{15}$.  
$(1)$ Find the value of $f(4)$;  
$(2)$ Calculate the value of $f(1)+f(2)+f(3)+f(4)+f(5)+f(6)+f(7)$;  
$(3)$ Calculate the value of $f(1)+f(2)+\ldots+f(n)$ (where $n$ is a positive integer).

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