
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
Live streaming e-commerce has become a shopping trend among young people. To thank his fans, the live streaming e-commerce influencer, Da Yang Ge, has introduced a promotion: fans can receive a $10\%$ subsidy at the online customer service center when purchasing leather jackets and sweaters in the live broadcast. Fan Li Li bought a leather jacket and a sweater in the live broadcast, spending a total of $3000. It is known that the price of a leather jacket is $600 more than 5 times the price of a sweater.<br/>$(1)$ What are the prices of the leather jacket and the sweater purchased by Li Li? How much subsidy can Li Li receive at the online customer service center?<br/>$(2)$ On that day, Da Yang Ge sold a total of 300 pieces of leather jackets and sweaters. In order to ensure that the total subsidy received at the online customer service center does not exceed $50000, how many sweaters should be sold at least?

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