
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 $10^{a}=b$, then $a$ is called the logarithm of $b$, denoted as $a=f\left(b\right)$. It is known from the definition that $10^{a}=b$ and $a=f\left(b\right)$ represent the same relationship between the two quantities $a$ and $b$.  
$(1)$ According to the definition of logarithm, fill in the blank: $f(10^{-2})=$______;  
$(2)$ Logarithms have the following properties:  
If $m$ and $n$ are positive numbers, then $f\left(mn\right)=f\left(m\right)+f\left(n\right)$ and $f(\frac{m}{n})=f(m)-f(n)$. According to these properties, fill in the blank: $\frac{f(b^3)}{f(b)}=\_\_\_\_\_\_$ (where $b$ is a positive number). If $f\left(2\right)\approx 0.3$, then $f\left(20\right)\approx$______, $f(\frac{1}{50})\approx$______ (answers are rounded to one decimal place).  
$(3)$ Given $f\left(3\right)=a$, $f\left(7\right)=b$, $f\left(0.63\right)=c$, the equal relationship between $a$, $b$, and $c$ is______.

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