
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 shopping mall needs $7400$ yuan to purchase $2$ units of type A mobile phones and $3$ units of type B mobile phones; it needs $11700$ yuan to purchase $3$ units of type A mobile phones and $5$ units of type B mobile phones.<br/>$(1)$ Find the cost price of each unit of type A and type B mobile phones;<br/>$(2)$ The shopping mall plans to purchase a total of $30$ units of these two types of mobile phones using funds of at least $44400$ yuan and at most $50000$ yuan.<br/>① Find out how many purchasing schemes are there;<br/>② If the selling prices of each unit of type A and type B mobile phones are $2500$ yuan and $1950$ yuan respectively, and the purchasing scheme with the least amount of type A mobile phones is chosen for purchase. In order to promote sales, the shopping mall decides to refund customers $a$ yuan for each unit of type A mobile phone sold and $b$ yuan for each unit of type B mobile phone sold $(a\geqslant 50$, $b\geqslant 50$, and $a$ and $b$ are multiples of $50$). To ensure a profit of $16500$ yuan from this purchasing scheme (selling all units), write down the values of $a$ and $b$.

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