
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
In an acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{sin(A-B)}{cosB}=\frac{sin(A-C)}{cosC}$.  
$(1)$ If angle $A=\frac{π}{3}$, find angle $B$;  
$(2)$ If $a\sin C=1$, find the maximum value of $\frac{1}{a^2}+\frac{1}{b^2}$.

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