
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
Fruit vendor Lao Xu purchases goods at the fruit wholesale market. He knows that the wholesale price of strawberries is $60$ yuan per box, and the wholesale price of apples is $40$ yuan per box. Lao Xu bought a total of $60$ boxes of strawberries and apples, spending exactly $3100$ yuan.<br/>$(1)$ How many boxes of strawberries and apples did he buy respectively?<br/>$(2)$ Lao Xu has two shops, Shop A and Shop B. For each box of strawberries and apples sold, Shop A makes a profit of $15$ yuan and $20$ yuan respectively, while Shop B makes a profit of $12$ yuan and $16$ yuan respectively. Lao Xu distributes the 60 boxes of fruit he purchased to Shop A with $a$ boxes of strawberries and $b$ boxes of apples, and the rest are distributed to Shop B. Due to his good reputation, both shops quickly sold out of this batch of fruit.<br/>① If Lao Xu makes a profit of $600$ yuan at Shop A, how much profit does he make at Shop B?<br/>② If Lao Xu wishes to make a total profit of $1000$ yuan, then $a+b=\_\_\_\_\_\_$. (Write down the answer directly)

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