For a positive real number $x$, let $[x]$ denote its integer part. For example, $[2.711] = 2$, $[7] = 7$, $[6.9] = 6$. 

Find the maximum real number $z$ such that $\left[\frac{5}{z}\right] + \left[\frac{6}{z}\right] = 7$. 

Calculate the value of $20z$.

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