11. (12 points) A store has 89 gold coins waiting to be sold, numbered from $1 \sim 89$, each priced at 30 yuan, and only one of them is a “lucky coin”. Feifei can choose some of the numbers and ask the honest clerk: “Is the lucky coin’s number among them?”. If the answer is “yes”, Feifei needs to pay 20 yuan for the consultation fee; if the answer is “no”, Feifei needs to pay 10 yuan for the consultation fee. Of course, she can also choose not to ask any questions and directly buy some of the coins. Feifei needs to pay at least $\qquad$ yuan to ensure she can definitely get the lucky coin.
【Analysis】Analysis: Consider a more general case: there are $n$ gold coins for sale, and only one of them is a lucky coin.
（1）Even when $n=2$, if Feifei does not ask any questions and directly buys the two gold coins, she will of course get the lucky coin, but she will need to pay 60 yuan in total; if Feifei asks about a certain number, to ensure she gets the gold coin, she needs to pay $20+30=50$ yuan. This means that no matter what $n$ is, Feifei needs to clearly identify which one is the lucky coin before making a purchase to be the most cost-effective. Therefore: The money Feifei needs to pay = the consulting fee for identifying the lucky coin + the purchase fee for buying the lucky coin
（2）Next, we explore the pattern to find out: the consulting fee $m$ that Feifei needs to pay to identify the lucky coin from $n$ gold coins for sale (Note: it either remains unchanged or increases by 10)
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline \begin{tabular}{l} 
Gold \\
Coins $n$ 
\end{tabular} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & $\cdots$ \\
\hline \begin{tabular}{l} 
Consulting \\
Fee $m$ yuan
\end{tabular} & 20 & 30 & 40 & 40 & 50 & 50 & 50 & $\cdots$ \\
\hline
\end{tabular}

By finding the pattern, we discover:
$2 \times 10$ yuan consulting fee, can identify the lucky coin from at most 2 gold coins, denoted as $E$ (2) $=2$; $3 \times 10$ yuan consulting fee, can identify the lucky coin from at most 3 gold coins, denoted as $E$ (3) $=3$; $4 \times 10$ yuan consulting fee, can identify the lucky coin from at most 5 gold coins, denoted as $E$ (4) $=5$; $5 \times 10$ yuan consulting fee, can identify the lucky coin from at most 8 gold coins, denoted as $E \quad(5)=8$;

Listing $2,3,5,8$ separately, we find the characteristic of this sequence: starting from the 3rd term, each term equals the sum of the two preceding terms. If we continue this sequence, it becomes:
$2, 3, 5, 8, 13, 21, 34, 55, 89 \cdots$ Up to this point, the answer is: consulting fee 100 yuan + purchase fee 30 yuan $=130$ yuan.
（3）Below is an explanation of the pattern discovered in (2):
We re-examine the meaning of $E(m)$: a consulting fee of $10 \mathrm{~m}$ yuan can identify the lucky coin from $E(m)$ or fewer gold coins, but cannot identify the lucky coin from more than $E(m)$ gold coins.

When $m \geqslant 3$, Feifei divides the gold coins into two piles and asks a question about the first pile.
If the answer is "yes," Feifei needs to pay 20 yuan first, and the remaining $10 \times(m-2)$ yuan consulting fee is used to identify the lucky coin from the first pile, which means the number of gold coins in the first pile cannot exceed $E(m-2)$; if the answer is "no," Feifei needs to pay 10 yuan first, and the remaining $10 \times(m-1)$ yuan consulting fee is used to identify the lucky coin from the second pile, which means the number of gold coins in the second pile cannot exceed $E(m-1)$ - so $E(m) \leqslant E(m-2) + E(m-1)$;

On the other hand, when there are $E(m-2) + E(m-1)$ gold coins for sale, Feifei divides the gold coins into the first pile of $E(m-2)$ and the second pile of $E(m-1)$, and then asks a question about the first pile, which means $10 m$ yuan can be used to identify the lucky coin from $E(m-2) + E(m-1)$.

Therefore, $E(m) = E(m-2) + E(m-1)$.
【Solution】Solution: From the analysis, we get the pattern
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|}
\hline \begin{tabular}{l} 
Gold \\
Coins $n$ 
\end{tabular} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & $\cdot$ & 89 & $\cdot$ \\
\hline \begin{tabular}{l} 
Consulting \\
Fee $m$ yuan
\end{tabular} & 20 & 30 & 40 & 40 & 50 & 50 & 50 & $\cdot$ & 100 & $\cdot$ \\
\hline
\end{tabular}
$100 + 30 = 130$ (yuan)
Therefore: The answer is 130.