
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
A right circular cone has base radius 1 cm and slant height 3 cm is given.   $P$  is a point on the circumference of the base and the shortest path from  $P$  around the cone and back to  $P$  is drawn (see diagram).  What is the minimum distance from the vertex  $V$  to this path?

[asy]
import graph;

unitsize(1 cm);

filldraw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(Circle((0,0),1)),gray(0.9),nullpen);
draw(yscale(0.3)*(arc((0,0),1.5,0,180)),dashed);
draw(yscale(0.3)*(arc((0,0),1.5,180,360)));
draw((1.5,0)--(0,4)--(-1.5,0));
draw((0,0)--(1.5,0),Arrows);
draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows);
draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,0,180)),dashed);
draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,180,360)));

label(" $V$ ", (0,4), N);
label("1 cm", (0.75,-0.5), N);
label(" $P$ ", (-1.5,0), SW);
label("3 cm", (1.7,2));
[/asy]

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