
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  $f:(0,\infty) \rightarrow (0,\infty)$  be a function such that 
\[ 
10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90
\]
for every  $x,y \in (0,\infty)$ . Evaluate $f(\frac 1{11})$.

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