
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 positive integer  $k,$  we say that it is a *Taurus integer* if  we can delete one element from the set  $M_k=\{1,2,\cdots,k\},$  such that the sum of remaining  $k-1$  elements is a positive perfect square. For example,  $7$  is a Taurus integer, because if we delete  $3$  from  $M_7=\{1,2,3,4,5,6,7\},$  the sum of remaining  $6$  elements is  $25,$  which is a positive perfect square. $(1)$  Determine whether  $2021$  is a Taurus integer. $(2)$  For positive integer  $n,$  determine the number of Taurus integers in  $\{1,2,\cdots,n\}.$ 

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