
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
Li's family is renovating their living room and needs a total of 100 tiles of a certain type. After a market survey, it was found that if they buy 40 colored tiles and 60 single-colored tiles, they would spend 5600 yuan. If they buy 50 colored tiles and 50 single-colored tiles, they would spend 6000 yuan.  
$(1)$ Find the unit price of each type of tile in yuan per tile.  
$(2)$ If they also need to lay these two types of tiles in the kitchen, a total of 60 tiles, and the cost of buying the tiles does not exceed 3400 yuan, how many colored tiles can they purchase at most?

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