
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
Two circles  $A$  and  $B$ , both with radius  $1$ , touch each other externally. 
Four circles  $P, Q, R$  and  $S$ , all four with the same radius  $r$ , lie such that $P$  externally touches on  $A, B, Q$  and  $S$ , $Q$  externally touches on  $P, B$  and  $R$ , $R$  externally touches on  $A, B, Q$  and  $S$ ,  $S$  externally touches on  $P, A$  and  $R$ . 
Calculate the length of  $r.$ [asy]
unitsize(0.3 cm);

pair A, B, P, Q, R, S;
real r = (3 + sqrt(17))/2;

A = (-1,0);
B = (1,0);
P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));
R = -P;
Q = (r + 2,0);
S = (-r - 2,0);

draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(P,r));
draw(Circle(Q,r));
draw(Circle(R,r));
draw(Circle(S,r));

label(" $A$ ", A);
label(" $B$ ", B);
label(" $P$ ", P);
label(" $Q$ ", Q);
label(" $R$ ", R);
label(" $S$ ", S);
[/asy]

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