
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 certain village, in promoting the activity of building beautiful countryside, decided to construct a Happiness Square and planned to lay red and blue tiles of the same size. After investigation, the following information was obtained:

|  | Quantity purchased less than $5000$ pieces | Quantity purchased not less than $5000$ pieces |
| --- | --- | --- |
| Red tiles | Sold at original price | Sold at 20% discount |
| Blue tiles | Sold at original price | Sold at 10% discount |

If $4000$ pieces of red tiles and $6000$ pieces of blue tiles are purchased, the payment is $86000$ yuan; if $10000$ pieces of red tiles and $3500$ pieces of blue tiles are purchased, the payment is $99000$ yuan. 

(1) What is the unit price of red tiles and blue tiles, respectively?
(2) After calculation, it is determined that $12000$ tiles need to be purchased, with the quantity of blue tiles not less than half of the quantity of red tiles and not exceeding $6000$ pieces. How to make the purchase with the minimum payment? Please explain the reason.

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