On a different SMP $4-$processor system, each of the four cores has a separate L$1$ cache, but the $4$ processors share a single L$2$ cache.$$ This system executes a different program that contains $4$ independent parts which can execute in parallel.$$ One option for improving the performance of the system is to provide each core with its own L$2$ cache so that the system contains four separate L$2$ caches instead of a single shared L$2$ cache. Can the actual speedup obtained by this improvement exceed a speedup of $4?$ Explain your answer.