Bellman initially developed dynamic programming for discrete temporal systems during the early 1950s [6, 7]. Examine a Markov decision framework with state domain $\mathcal X$, action domain $\mathcal A$, transition mechanism $P(\cdot\mid x,a)$, reward mapping $r(x,a)$, and discount parameter $\gamma\in(0,1)$. A strategy $\pi$ associates states with action distributions. Given state evolution as a controlled Markov chain
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