
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
It is said that our famous mathematician, Hua Luogeng, saw a puzzle in a magazine that a passenger next to him was reading while on a trip abroad. The puzzle was: a number is $59319$, and the task is to find its cube root. Hua Luogeng blurted out: $39$. The passenger next to him was very surprised and asked about the calculation's secret. Do you know how Hua Luogeng calculated it quickly and accurately? Please try the following questions:<br/>$(1)$ Given $10^{3}=1000$ and $100^{3}=1000000$, how did you determine how many digits $\sqrt[3]{{59319}}$ has?<br/>$(2)$ Given that the unit digit of $59319$ is $9$, how did you determine what the unit digit of $59319$ is?<br/>$(3)$ If you subtract the last three digits $319$ from $59319$ and get the number $59$, and $3^{3}=27$, $4^{3}=64$, how did you determine the tens digit of $\sqrt[3]{{59319}}$?<br/>$(4)$ Given that $6859$, $19683$, and $110592$ are all cubes of integers, determine their cube roots using the above method (write down the results directly).

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