
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 conical pendulum is formed from a rope of length  $ 0.50 \, \text{m} $  and negligible mass, which is suspended from a fixed pivot attached to the ceiling. A ping-pong ball of mass  $ 3.0 \, \text{g} $  is attached to the lower end of the rope. The ball moves in a circle with constant speed in the horizontal plane and the ball goes through one revolution in  $ 1.0 \, \text{s} $ . How high is the ceiling in comparison to the horizontal plane in which the ball revolves? Express your answer to two significant digits, in cm. 

*(Proposed by Ahaan Rungta)*

<details><summary>Clarification</summary>During the WOOT Contest, contestants wondered what exactly a conical pendulum looked like. Since contestants were not permitted to look up information during the contest, we posted this diagram: [asy]
size(6cm);
import olympiad;
draw((-1,3)--(1,3));
draw(xscale(4) * scale(0.5) * unitcircle, dotted);
draw(origin--(0,3), dashed);
label(" $h$ ", (0,1.5), dir(180));
draw((0,3)--(2,0));
filldraw(shift(2) * scale(0.2) * unitcircle, 1.4*grey, black);
dot(origin);
dot((0,3));[/asy]The question is to find  $h$ .</details>

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