
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
In February, cherry blossoms bloom, and in April, cherries turn red. As cherries ripen and hit the market, a certain fruit shop spends $6000$ yuan to purchase yellow honey cherries and another $1000$ yuan to purchase red light cherries. The cost price of yellow honey cherries is twice that of red light cherries, and the quantity of yellow honey cherries is 100 kilograms more than the quantity of red light cherries.  
$(1)$ Find the cost price per kilogram of red light cherries.  
$(2)$ In the first week, the fruit shop sells red light cherries at a price of $40$ yuan per kilogram for $3m$ kilograms. In the second week, the selling price per kilogram decreases by $0.5m$ yuan, and $20$ kilograms are sold. In the third week, the selling price is discounted by 30% based on the first week's price. The remaining red light cherries are sold out. If the total profit from the red light cherries purchased is not less than $770$ yuan, find the minimum value of $m$.

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