
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
Define a new operation: $a*b=\frac{1}{a}-\frac{1}{b}$, $a⊗b=\frac{1}{ab}$. For example, $3*7=\frac{1}{3}-\frac{1}{7}=\frac{4}{21}$, $3⊗7=\frac{1}{3×7}=\frac{1}{21}$. If $a\otimes b=a*b$, then the rational numbers $a$ and $b$ are called a "happy number pair". For example, $2⊗3=\frac{1}{2×3}=\frac{1}{6}$, $2*3=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$, $2\otimes 3=2*3$, so $2$ and $3$ are a "happy number pair".

$(1)$ The following groups of numbers are "happy number pairs"______; (please fill in the numbers)<br/>① $a=1$, $b=2$; ② $a=-1$, $b=1$; ③ $a=-\frac{4}{3}$, $b=-\frac{1}{3}$.<br/>$(2)$ Calculate: $\left(-3\right)*4-\left(-3\right)\otimes 4+\left(-31415\right)*\left(-31415\right)$;<br/>$(3)$ Given that two consecutive non-zero integers form a "happy number pair", calculate: $1\otimes 2+2\otimes 3+3\otimes 4+4\otimes 5+\cdots +2021\otimes 2022$.

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