
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 swimming pool offers two payment methods for swimming:<br/>Method 1: Purchase a membership card for $200$ yuan each, which can only be used by the cardholder in the current year. Each swim session with the card requires an additional payment of $10$ yuan;<br/>Method 2: Without purchasing a membership card, each swim session costs $30$ yuan.<br/>$(1)$ If swimming $3$ times using Method 1 for payment, the total cost is ______ yuan;<br/>$(2)$ If swimming $9$ times, which payment method is more cost-effective? Please explain through calculations;<br/>$(3)$ If Xiaoxuan expects his swimming expenses to be $600$ yuan this year, which payment method allows for more swimming sessions? Please explain.

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