For each $t\geqslant0$, let $S_{t}$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$ f(1)-2 f(2 / 3)+f(1 / 3)\geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] . $$ Develop necessary and sufficient conditions on $t$ for $S_{t}$ to be closed under multiplication. Please provide the sum of the endpoints of the interval for which the condition holds.

Let's think step by step and output the final answer within \boxed{}.
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