
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A software company has developed a new game software, which sets several levels of games according to the difficulty of the game. To stimulate the enthusiasm for passing levels, a certain number of Hui coins (a type of virtual currency) are awarded for passing each level. Let the reward for passing the $n$-th level be $a_n$ Hui coins, and it satisfies $\frac{1}{2}a_n \leqslant a_{n+1} \leqslant 4a_n$, with $a_1=1$. The software offers two reward schemes: the first scheme, starting from the second level, the number of Hui coins awarded for passing each level is $q$ times that of the previous level; the second scheme, starting from the second level, each level awards $d$ more Hui coins than the previous level ($d\in\mathbb{R}$). The game rules require the player to choose one of the reward schemes before starting.  
(Ⅰ) If the first scheme is chosen, let the total number of Hui coins awarded from the first level to the $n$-th level be $S_n$, i.e., $S_n=a_1+a_2+\ldots+a_n$, and $\frac{1}{2}S_n \leqslant S_{n+1} \leqslant 4S_n$, find the range of values for $q$;  
(Ⅱ) If the second scheme is chosen, and the total number of coins awarded from the first level to the $k$-th level is set to $100$ (i.e., $a_1+a_2+a_3+\ldots+a_k=100$, $k\in\mathbb{N}^*$) to receive a special award, how should the value of $d$ be set to maximize $k$, and find the maximum value of $k$.

Present the answer in LaTex format: \boxed{Your answer}
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