For each of the following assertions, support your answer with examples or counterexamples:$1$For any local$-$search problem, hill$-$climbing will return a global optimum if the algorithm is run starting at any state that is a neighbor of a neighbor of a globally optimal state$\mathrm{.}2$In a nondeterministic, partially observable problem, improving an agents transition model $($i$.$e$.,$ eliminating spurious outcomes that never actually occur$)$ will never increase the size of the agents belief states$\mathrm{.}3$Stochastic hill climbing is guaranteed to arrive at a global optimum$\mathrm{.}4$In a continuous space, gradient descent with a fixed step size is guaranteed to converge to a local or global optimum$\mathrm{.}5$Gradient descent finds the global optimum from any starting point if and only if the function is convex.$$