
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
Please read the following dialogue and answer the questions:<br/>One evening after dinner, Xiaoming chatted with the owner of the neighboring small shop. The shop owner said: I deal with two types of products, $A$ and $B$. The sum of the purchase prices of products $A$ and $B$ is $5$ yuan; the retail price of product $A$ is $1$ yuan more than the purchase price, and the retail price of product $B$ is $1$ yuan less than twice the purchase price. Buying $3$ pieces of product $A$ and $2$ pieces of product $B$ at the retail price costs a total of $19$ yuan. Do you know the purchase prices of products $A$ and $B$? Xiaoming quickly answered the shop owner's question after thinking for a while. He then asked the shop owner a question: Last time I went shopping at the supermarket and bought three items, $A$, $B$, and $C. I initially took $4$ pieces of item $A$, $7$ pieces of item $B$, and $1$ piece of item $C, and the cashier told me it would cost $8$ yuan. I didn't have enough money, so I changed my purchase to $2$ pieces of item $A$, $3$ pieces of item $B$, and $1$ piece of item $C, and the cashier told me it would cost $6$ yuan. Still, I didn't have enough money. After calculating, I found that I had just enough money to buy one piece of each item, $A$, $B$, and $C. Do you know how much money I brought that day? The shop owner was stunned and sighed: I have no idea! The younger generation is really impressive!<br/>Questions:<br/>$(1)$ Do you know how Xiaoming solved the shop owner's question? Please write out the solution process.<br/>$(2)$ Can Xiaoming's question to the shop owner really not be solved? If it can be solved, please write out the solution process.

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