
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
Circle  $B$ , which has radius 2008, is tangent to horizontal line  $A$  at point  $P$ .  Circle  $C_1$  has radius 1 and is tangent both to circle  $B$  and to line  $A$  at a point to the right of point  $P$ .  Circle  $C_2$  has radius larger than 1 and is tangent to line  $A$  and both circles B and  $C_1$ . For  $n>1$ , circle  $C_n$  is tangent to line  $A$  and both circles  $B$  and  $C_{n-1}$ . Find the largest value of n such that this sequence of circles can be constructed through circle  $C_n$  where the n circles are all tangent to line  $A$  at points to the right of  $P$ .

[asy]
size(300);
draw((-10,0)--(10,0));
draw(arc((0,10),10,210,330));
label(" $P$ ",(0,0),S);
pair C=(0,10),X=(12,3);
for(int kk=0;kk<6;++kk)
{
pair Y=(X.x-X.y,X.y);
for(int k=0;k<20;++k)
Y+=(abs(Y-X)-X.y-Y.y,abs(Y-C)-10-Y.y)/3;
draw(circle(Y,Y.y));
X=Y;
}[/asy]

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