
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
Observe the equation:

$$
\begin{aligned}
(1+2+3+4)^{2} & =(1+2+3+4)(1+2+3+4) \\
& =1 \cdot 1+1 \cdot 2+1 \cdot 3+1 \cdot 4+2 \cdot 1+2 \cdot 2+2 \cdot 3+2 \cdot 4+ \\
& +3 \cdot 1+3 \cdot 2+3 \cdot 3+3 \cdot 4+4 \cdot 1+4 \cdot 2+4 \cdot 3+4 \cdot 4
\end{aligned}
$$

Note that \(4 \times 4=16\) products are formed when calculating \((1+2+3+4)^{2}\) using the distributive property.

a) How many products will be formed when calculating \((1+2+3+4)^{3}\) also using the distributive property?

b) What is the quantity of two-digit numbers that use only the digits 1, 2, 3, and 4?

c) What is the sum of the products of the digits of all four-digit numbers formed using only the digits 1, 2, 3, and 4?

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