We are given n key$-$value pairs $($you may assume that all keys and values are distinct$),$ stored in a binary search tree T$.$ Each node of the tree stores $3$ pointers: left$-$child, right$-$child, and parent. Each node of the tree also stores the key and value of the element stored there. You may assume that the tree is balanced $-$ such as an AVL$-$tree, so that the height of the tree is $\backslash$Theta $($lg n$).$ Augment the tree to support the following operations in O$($lg n$)$ time. For each operation, describe how the tree would be augmented, and how the operation would be implemented.

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$($a$)$ Max$-$Value$-$Suffix$($T$,$k$)$: return the maximum value among all key$-$value pairs with key greater than or equal to k$.$

$($b$)$ Filter$-$Min$-$Key$($T$,$v$)$: return the minimum key value over all key$-$value pairs with the value at most v$.$