
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 tour group of 33 people stayed at a hotel. The hotel had only 4 single rooms and several triple rooms left. The price for accommodation was 100 yuan per night for a single room and 130 yuan per night for a triple room. (Note: Men can only stay with men, and women can only stay with women. The triple rooms do not need to be fully occupied, but the price of 130 yuan per night still applies for each room.)<br/>$(1)$ If the accommodation fee for one night for this tour group was 1530 yuan, then they rented ____ single rooms;<br/>$(2)$ If the tour group rented 3 single rooms and there were a total of 19 men, then the minimum cost for one night of accommodation would be ____ yuan.

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