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You have run through the simulations. Hundreds, thousands of your best simulation models endlessly test and refine the power process you have theorized. In comparison to the old way, all four of your new models form a <span class="mu-i"><span class="mu-s">roughly equal level of stability and chance of long term success-</span></span> or failure. While some things in society will have to change to accommodate whichever method of selection you choose; you are confident that the changes to the new style of regime will be mostly painless.
You gather your most influential advisors, department leaders, Overseers, and high ranking state philosophers to lay out your newest plans. You explain the problem, and then the solutions, and they seem to agree. Bluey is the most enthusiastic, of course, but to him you can do no wrong. Your brother, Kimnan II, is also willing to accept the deep changes you are making to the Hegemony. Everyone is willing and ready to complete this transition without complaint- and they are confident this will not have any serious blowback in the general population, or even among alien minorities, though these groups already lack any serious sway or ability to complain in the first place.
<span class="mu-i"><span class="mu-s">Whichever new method of choosing Supreme Rulers you decide</span></span>, they are all inherently as stable as each other, and each will be able to continue your traditions as best as possible. None will inherently cause a succession crisis or public uprising. All will be equal, <span class="mu-s"><span class="mu-i">the only difference being what Supreme Rulers</span></span> they will create. All have an equal chance of being subverted, either from inner and outer forces, and working on defending the institution is an evolving struggle all will face evenly.
For this deep and fundamental task; you have created four methods. While there is always room for refinement and tweaking as time goes on; One of these which you will finally choose and, from this point on and forever, the <span class="mu-s"><span class="mu-i">Hegemony will be run with this style of succession</span></span>.