We are trying to find a hidden number selected from the set of integers $\{1,2,\dots ,n\}$ by asking questions. Each number (question) we ask, we get one of three possible answers:

• "Your guess is lower than the hidden number" (and you incur a cost of $a$), or
• "Your guess is higher than the hidden number" (and you incur a cost of $b$), or
• "Yes, that's it!" (and the game ends).

Given the value of $n$, $a$, and $b$, an optimal strategy minimizes the total cost for the worst possible case.

For example, if $n=5$, $a=2$, and $b=3$, then we may begin by asking "2" as our first question.

If we are told that 2 is higher than the hidden number (for a cost of b=3), then we are sure that "1" is the hidden number (for a total cost of 3).
If we are told that 2 is lower than the hidden number (for a cost of a=2), then our next question will be "4".
If we are told that 4 is higher than the hidden number (for a cost of b=3), then we are sure that "3" is the hidden number (for a total cost of 2+3=5).
If we are told that 4 is lower than the hidden number (for a cost of a=2), then we are sure that "5" is the hidden number (for a total cost of 2+2=4).
Thus, the worst-case cost achieved by this strategy is 5. It can also be shown that this is the lowest worst-case cost that can be achieved. So, in fact, we have just described an optimal strategy for the given values of $n$, $a$, and $b$.

Let $C(n,a,b)$ be the worst-case cost achieved by an optimal strategy for the given values of $n$, $a$ and $b$.

Here are a few examples:
$C(5,2,3)=5$
$C(500,\sqrt{2},\sqrt{3})=13.22073197\dots$
$C(20000,5,7)=82$
$C(2000000,\sqrt{5},\sqrt{7})=49.63755955\dots$

Let $F_k$ be the Fibonacci numbers: $F_k=F_{k-1}+F_{k-2}$ with base cases $F_1=F_2=1$.
Find ${\displaystyle \sum _{k=1}^{30}C({10}^{12},\sqrt{k},\sqrt{F_k})}$, and give your answer rounded to 8 decimal places behind the decimal point.