
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  $x,$  let  $\lfloor x\rfloor$  denote the greatest integer less than or equal to  $x,$  and let  $\{x\} = x -\lfloor x\rfloor$  denote the fractional part of  $x.$  The sum of all real numbers  $\alpha$  that satisfy the equation   $$ \alpha^2+\{\alpha\}=21 $$   can be expressed in the form   $$ \frac{\sqrt{a}-\sqrt{b}}{c}-d $$   where  $a, b, c,$  and  $d$  are positive integers, and  $a$  and  $b$  are not divisible by the square of any prime. Compute  $a + b + c + d.$ 

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