Problem Description:

Being a some constant, further assume that we
are in a factor ring (basically all operations modulo some sumber p). Note, that the division below is a multiplication by the modular inverse.
You always have to start with x=9.
Consider the following recursive formula:


How often do you have to perform this operation to get a specific x (basically getting the new_x and feeding it back into the formula to get another new_x, and so on)?
Note: You can start multiple such chains beginning at x=9, and add the resulting x values
using the addition algorithm from http://en.wikipedia.org/wiki/Montgomery_curve (Montgomery arithmetic section).
Note, that the x value, is the value you get at the end of such calculation-chain, and the z value is always 1.

The winner:
The winner is the first person to post such formula in private. The formula must work in all cases, and be comutationally feasible (let us say calculatable in less than 24 hours).
If there are any wrong descriptions in this post at this time, they may be corrected or adjusted later on. Bounty ends on 02/15/2015.