
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
For a sequence of number pairs P: $(a_1, b_1)$, $(a_2, b_2)$, ..., $(a_n, b_n)$, where $(a_i, b_i \in \mathbb{R}^+, i=1, 2, 3, ..., n)$, define $f_0(y) = 0$ (for $y \geq 0$) and $f_k(y) = \max_{x_k=0,1,2,3,\ldots,m} \{b_kx_k + f_{k-1}(y-a_kx_k)\}$ (for $y \geq 0$, $1 \leq k \leq n$), where m is the greatest integer not exceeding $\frac{y}{a_k}$. (Note: $\max_{x_k=0,1,2,3,\ldots,m} \{b_kx_k + f_{k-1}(y-a_kx_k)\}$ indicates the maximum number in the expression $b_kx_k + f_{k-1}(y-a_kx_k)$ when $x_k$ takes values $0, 1, 2, 3, \ldots, m$)

Given the sequence of number pairs P: $(2, 3)$, $(3, 4)$, $(3, p)$, answer the following questions:
(Ⅰ) Write down the value of $f_1(7)$;
(Ⅱ) Calculate the value of $f_2(7)$, and the corresponding values of $x_1$, $x_2$;
(Ⅲ) When the value of $f_3(11)$ is determined with $x_1=4$, $x_2=0$, $x_3=1$, write down the range of possible values for $p$ (only the result is required, no need to explain the reasoning).

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