Consider two binary search trees $($BSTs$)$ named T$1$ and T$2,$ rooted at Root$1$ and Root$2,$ respectively. Both T$1$ and T$2$ are AVL $($Adelson$-$Velsky$-$Landis$)$ trees with M$(>0)$ and N$(>0)$ nodes, respectively. Every node of T$1$ and T$2$ has three fields: value, left$-$link and right$-$link. T$1$ and T$2$ have no common value. Define merging of a BST T$2$ with a BST T$1$ as a process that combines the two BSTs to form a single BST by attaching the root of T$2$ to a left$-$ or right$-$link of T$1$ that is free. Thus, the merge operation excludes one by one insertion of the nodes of one BST into the other. $($a$)$ Create two AVL trees TA $($with $6$ integer values$)$ and TB $($with $7$ integer values$),$ such that TB cannot be merged with TA as per the above definition. Note that TA and TB must not have any common value. $($b$)$ Design an algorithm that takes two AVL trees T$1$ and T$2$ as input and merges T$2$ with T$1,$ if possible, otherwise reports an error. Explain your algorithm. Deduce the time complexity of your algorithm, and also find the maximum possible height of the merged tree in terms of the heights of T$1$ and T$2.$