
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
If the numerator or denominator of a fraction can be factorized, and the fraction cannot be reduced, then we call this fraction a "harmonic fraction."  
$(1)$ The following fractions: ① $\frac{x-1}{{{x^2}+1}}$; ② $\frac{a-2b}{{{a^2}-{b^2}}}$; ③ $\frac{x+y}{{{x^2}-{y^2}}}$; ④ $\frac{{{a^2}-{b^2}}}{{{{(a+b)}^2}}}$. Among them, the "harmonic fractions" are ______ (just fill in the numbers);  
$(2)$ If $a$ is a positive integer, and $\frac{x-1}{{{x^2}+ax+4}}$ is a "harmonic fraction," please write down the value of $a$;  
$(3)$ When simplifying $\frac{{4{a^2}}}{{a{b^2}-{b^3}}}-\frac{a}{b}÷\frac{b}{4}$,  
Xiaodong and Xiaoqiang respectively carried out the following three steps:  
Xiaodong: $Original = \frac{{4{a^2}}}{{a{b^2}-{b^3}}}-\frac{a}{b}×\frac{4}{b}=\frac{{4{a^2}}}{{a{b^2}-{b^3}}}-\frac{4a}{b^2}=\frac{{4{a^2}{b^2}-4a({a{b^2}-{b^3}})}}{{({a{b^2}-{b^3}}){b^2}}}$  
Xiaoqiang: $Original = \frac{{4{a^2}}}{{a{b^2}-{b^3}}}-\frac{a}{b}×\frac{4}{b}=\frac{{4{a^2}}}{{{b^2}({a-b})}}-\frac{4a}{b^2}=\frac{{4{a^2}-4a({a-b})}}{{({a-b}){b^2}}}$  
Obviously, Xiaoqiang used the harmonic fraction in it. The result obtained in the third step is simpler than that of Xiaodong. The reason is: ______,  
Please continue with Xiaoqiang's method to complete the simplification.

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