a$)$ Draw a Max Heap tree using the following array of elements that includes key $($integer$)$ and data$-$value $($string$)$:$$

$[(15,$Ali$),(25,$Dav$),(10,$Bob$),(17,$Mat$),(30,$Wil$),(45,$Mad$),(40,$Ann$),(37,$Alex$)]$

Write down the steps you followed to construct the Max Heap tree from the given array. Ensure that the tree structure adheres to the properties of a Max Heap.

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b$)$ Consider the given Max Heap tree. Insert a new node with the key and data$-$value $(27,$ "Tar"$)$ into the Max Heap tree. Follow all rules of up$-$heap and down$-$heap as necessary to maintain the Max Heap property.$$

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c$)$ Remove the maximum node from the tree. After removing the minimum node, update the tree while following all down$-$heap rules as necessary to maintain the Max Heap property.