
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 a function $f(x)$ such that $f(x)=\frac{1}{x-1}$ for $x\neq 1$.  
$(1)$ Find the analytical expression for $f(2-x)$.  
$(2)$ Find the value of $f(\frac{1}{20})+f(\frac{3}{20})+f(\frac{5}{20})+\cdots+f(\frac{35}{20})+f(\frac{37}{20})+f(\frac{39}{20})$.

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