🚀 go-pugleaf

HERE IS A CHALLENGE DESIGN QUESTION.
ANYONE WITH A GOOD WORKABLE SOLUTION WILL BE WELL REWARDED.
PLEASE RESPOND DIRECTLY TO ME VIA EMAIL, AND PERHAPS ALSO WITH A POST

I have the following circuit:

                    (Switched Voltage) VS
                            |
                            R1
                            |
                       +----+---- (Output) VO
                       |    |
                       C    R3
                       |    |
                       +----+
                            |
                        (Ground)

The switched voltage VS is alternated between a reference, VR,
and ground.  When switched to VS, the circuit charges C and VO
rises.  When VO reaches 2/3 of VR, the circuit is automatically
switched to ground and begins to discharge C.  When C reaches
1/3 of VR, the circuit is automatically switched back to VR and
the cycle repeats.

The objective is to determine the value that VO would reach when
VS is switched to VR and C is missing (same as if infinite time
has passed).  Unfortunately, this value can not be measured directly.
Instead, I can only measure the time that it takes for VO to independently
charge or discharge between 1/3 and 2/3 of VR (VR known if necessary).

Assume that the values of R1 and R3 are such that VO actually
reaches 2/3 of VR, and in a quick time frame.

Approximations must be proven to be within better than 0.1%, otherwise
they should be avoided.