
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
During the Mid-Autumn Festival, a supermarket plans to introduce two types of mooncakes, Type A and Type B. They plan to spend $1200$ yuan on Type A mooncakes and $600$ yuan on Type B mooncakes. The cost of one Type A mooncake and one Type B mooncake is $9$ yuan in total, and the quantity of Type A mooncakes purchased is four times the quantity of Type B mooncakes.  
$(1)$ Find out how many Type A and Type B mooncakes they plan to purchase respectively.  
$(2)$ To thank the customers, the manufacturer has launched a series of promotions. The selling price of each Type A mooncake is reduced by $\frac{1}{3}$, and the selling price of each Type B mooncake is reduced by $\frac{m}{5}$ yuan (where $m\neq 0$). Now, based on the original plan in $(1)$, the quantity of Type B mooncakes purchased has increased by $\frac{15}{2}m$, but the total quantity of Type A and Type B mooncakes remains the same. The final total cost is reduced by $(400+2m)$ yuan compared to the original plan. Find the value of $m$.

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