Parameters and period of a cosine function from a maximum and a minimum, and an equivalent sine form

The function \(\mathrm{h}\) is defined by \(\mathrm{h}(x)=a\cos(bx)+3\) for \(x\in\mathbb{R}\), where \(a\) and \(b\) are real constants. Part of the graph of \(y=\mathrm{h}(x)\) is shown below. It has a local maximum point at \((0,7)\), and the next local minimum point to the right is \((6,-1)\).

1. Find the value of
   1. \(a\)
   2. \(b\), given that \(b>0\).[3]
2. Write down the least positive value of \(k\) for which \(\mathrm{h}(x+k)=\mathrm{h}(x)\) for every real \(x\).[1]
3. The function can also be expressed as \(\mathrm{h}(x)=p\sin\left(b(x-q)\right)+3\), where \(p\) and \(q\) are positive integers and \(b\) takes the value found in (a)(ii).
   1. Find the least possible value of \(q\).
   2. Write down the corresponding value of \(p\).[3]