
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
While running from an unrealistically rendered zombie, Willy Smithers runs into a vacant lot in the shape of a square,  $100$  meters on a side. Call the four corners of the lot corners  $1$ ,  $2$ ,  $3$ , and  $4$ , in clockwise order. For  $k = 1, 2, 3, 4$ , let  $d_k$  be the distance between Willy and corner  $k$ . Let

(a)  $d_1<d_2<d_4<d_3$ ,

(b)  $d_2$  is the arithmetic mean of  $d_1$  and  $d_3$ , and

(c)  $d_4$  is the geometric mean of  $d_2$  and  $d_3$ .

If  $d_1^2$  can be written in the form  $\dfrac{a-b\sqrt c}d$ , where  $a,b,c,$  and  $d$  are positive integers,  $c$  is square-free, and the greatest common divisor of  $a$ ,  $b$ , and  $d$  is  $1,$  find the remainder when  $a+b+c+d$  is divided by  $2008$ .

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