
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
Consider a sequence  $\{a_n\}_{n\geq 0}$  such that  $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$ .

(1) If  $a_0=24$ , then find the smallest  $n$  such that  $a_n=0$ .

(2) If  $a_0=m^2\ (m=2,\ 3,\ \cdots)$ , then for  $j$  with  $1\leq j\leq m$ , express  $a_{2j-1},\ a_{2j}$  in terms of  $j,\ m$ .

(3) Let  $m\geq 2$  be integer and for integer  $p$  with  $1\leq p\leq m-1$ , let  $a\0=m^2-p$ . Find  $k$  such that  $a_k=(m-p)^2$ , then 
find the smallest  $n$  such that  $a_n=0$ .

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