
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 in the summer. Method one: first purchase a membership card for $120$ yuan, which can only be used by the cardholder in the current year. Each time the cardholder swims, an additional $10$ yuan is charged. Method two: without purchasing a membership card, each swim costs $15$ yuan. Let Xiao Cong plan to swim $x$ times in the summer this year.<br/>$(1)$ According to the question, fill in the table below:<br/>

| Number of swims | $10$ | $15$ | $20$ | $\ldots $ | $x$ |
|------------------|------|------|------|-----------|-----|
| Total cost for method one (yuan) | $220$ | $270$ | ______ | $\ldots $ | ______ |
| Total cost for method two (yuan) | $150$ | $225$ | ______ | $\ldots $ | ______ |

$(2)$ If Xiao Cong plans to spend a total of $300$ yuan on swimming this summer, calculate and determine which payment method she should choose if she wants to swim more times.<br/>$(3)$ Teacher Zhang is a swimming enthusiast and plans to swim at least $40$ times at this swimming pool this summer. By calculation, determine which method is more cost-effective for Zhang.

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