
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
Given sets $A=\{x| x=4k+1, k\in Z\}$, $B=\{x| a\leqslant x < a+20, a\in R\}$, $C=\{x|\sqrt{x-5} < 5\}$.  
$(1)$ If $a=0$, find the number of elements in $A\cap B$;  
$(2)$ If $B\cap C\neq \varnothing $, find the range of values for $a$.

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