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A computer uses $46-bit$ virtual address, $32-bit$ physical address, and a three–level paged page table organization. The page table base register stores the base address of the first-level table ($T1$), which occupies exactly one page. Each entry of $T1$ stores the base address of a page of the second-level table ($T2$). Each entry of $T2$ stores the base address of a page of the third-level table ($T3$). Each entry of $T3$ stores a page table entry ($PTE$). The $PTE$ is $32$ bits in size. The processor used in the computer has a $1$ MB $16$ way set associative virtually indexed physically tagged cache. The cache block size is $64$ bytes.


What is the minimum number of page colours needed to guarantee that no two synonyms map to different sets in the processor cache of this computer?

  1. $2$
  2. $4$
  3. $8$
  4. $16$
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