
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
For an integer $n$, define $[\sqrt{n}]$ as the largest integer not greater than $\sqrt{n}$, for example: $[\sqrt{3}]=1$, $[\sqrt{4}]=2$, $[\sqrt{5}]=2$.  
$(1)$ Write down the value of $[\sqrt{10}]$ directly.  
$(2)$ Obviously, when $[\sqrt{n}]=1$, $n=1$, $2$, or $3$.  
① When $[\sqrt{n}]=2$, write down directly the values of $n$ that satisfy the condition.  
② When $[\sqrt{n}]=10$, find the number of $n$ that satisfy the condition.  
$(3)$ Perform the following operations on $72$: $72\stackrel{1st}{→}[\sqrt{72}]=8\stackrel{2nd}{→}[\sqrt{8}]=2\stackrel{3rd}{→}[\sqrt{2}]=1$, that is, after $3$ operations, $72$ becomes $1$. Similarly:  
① After how many operations does $25$ become $2$?  
② After $3$ operations, an integer $m$ becomes $2$, write down directly the maximum value of $m$.

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