
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 a real number $a$, we define: using the symbol $[\sqrt{a}]$ to represent the largest integer not greater than $\sqrt{a}$, and call $[\sqrt{a}]$ the root integer of $a$. For example: $[\sqrt{9}]=3$, $[\sqrt{10}]=3$, $[\sqrt{15}]=3$.  
$(1)$ Following the above method, calculate: $[\sqrt{25}]=\_\_\_\_\_\_$; $[\sqrt{50}]=\_\_\_\_\_\_$.  
If we continuously calculate the root integer of $a$ until the result is $1$. For example: for $10$, continuously calculate the root integer $2$ times, $[\sqrt{10}]=3→[\sqrt{3}]=1$, at this point the result is $1$.  
$(2)$ For $290$, how many times do we need to continuously calculate the root integer until the result is $1$?

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