
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 $(1-3x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots+a_{n}x^{n}$ where $n\in\mathbb{N}^{*}$.<br/>$(1)$ If $a_{2}=15a_{0}-13a_{1}$, find the value of $n$;<br/>$(2)$ If $n=2023$, $A=a_{0}+a_{2}+a_{4}+\cdots +a_{2022}$, $B=a_{1}+a_{3}+a_{5}+\cdots +a_{2023}$, find the value of $\frac{A-B}{A+B}$ (express the result in exponential form).

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