
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
Complete the table below, discover the patterns of square roots and cube roots, and apply the patterns to solve the problem.

| $x$     | $\ldots $ | $0.064$ | $0.64$ | $64$  | $6400$ | $64000$ | $\ldots $ |
|---------|-----------|---------|--------|-------|--------|---------|-----------|
| $\sqrt{x}$ | $\ldots $ | $0.25298$ | $0.8$ | $8$   | $m$    | $252.98$ | $\ldots $ |
| $\sqrt[3]{x}$ | $\ldots $ | $n$     | $0.8618$ | $4$   | $18.566$ | $40$    | $\ldots $ |

$(1)$ $m=$______, $n= \_\_\_\_\_\_.$  
$(2)$ From the numbers in the table, it can be observed that when finding the square root, if the decimal point of the radicand moves two places to the left (or right), the decimal point of its square root moves one place to the left (or right). Please describe in words the pattern of cube root: ______.  
$(3)$ If $\sqrt{a}≈14.142$, $\sqrt[3]{700}≈b$, find the value of $a+b$. (Reference data: $\sqrt{2}≈1.4142$, $\sqrt{20}≈4.4721$, $\sqrt[3]{7}≈1.9129$, $\sqrt[3]{0.7}≈0.8879$)

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