
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 2022, the Beijing Winter Olympics was successfully held, and the mascots and other souvenirs were highly popular. A certain store purchased two types of souvenirs, type A and type B, in equal quantities before the Winter Olympics. The costs for type A and type B were $10400$ yuan and $14000$ yuan, respectively. It is known that the cost price for each type B souvenir is $9$ yuan more than each type A souvenir.<br/>$(1)$ Find the cost price for each type of souvenir.<br/>$(2)$ During sales, it was found that when the selling price for each type A souvenir was $46$ yuan, 40 units were sold per day, and when the selling price for each type B souvenir was $45$ yuan, 80 units were sold per day. The store decided to reduce the price of type A souvenir and increase the price of type B souvenir. As a result, for every $1$ yuan reduction in the price of type A souvenir, 4 more units were sold, and for every $1$ yuan increase in the price of type B souvenir, 2 fewer units were sold. If on a certain day, a total of $140$ souvenirs were sold, what was the maximum profit made on that day?

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