
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
Let  $d$  and  $n$  be positive integers such that  $d$  divides  $n$ ,  $n > 1000$ , and  $n$  is not a perfect square.  The minimum possible value of  $\left\lvert d - \sqrt{n} \right\rvert$  can be written in the form  $a\sqrt{b} + c$ , where  $b$  is a positive integer not divisible by the square of any prime, and  $a$  and  $c$  are nonzero integers (not necessarily positive).  Compute  $a+b+c$ .

*Proposed by Matthew Lerner-Brecher*

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