
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 amusement park sells regular tickets for $40$ yuan each. To promote sales, they have introduced two new card options:<br/>Option 1: Platinum card priced at $200$ yuan each, with an additional charge of $20$ yuan each time;<br/>Option 2: Diamond card priced at $1000$ yuan each, with no additional charges each time.<br/>$(1)$ Fill in the table according to the question:<br/><table><tbody><tr><td>Number of visits to the amusement park</td><td>$10$</td><td>$20$</td><td>$50$</td><td>$\ldots $</td><td>$x\left(x \gt 0\right)$</td></tr><tr><td>Cost with regular tickets (yuan)</td><td>$400$</td><td>$800$</td><td>______</td><td>$\ldots $</td><td>______</td></tr><tr><td>Cost with Option 1 (yuan)</td><td>$400$</td><td>______</td><td>______</td><td>$\ldots $</td><td>______</td></tr><tr><td>Cost with Option 2 (yuan)</td><td>$1000$</td><td>$1000$</td><td>$1000$</td><td>$\ldots $</td><td>$1000$</td></tr></tbody></table>$(2)$ If Xiao Hong plans to spend $680$ yuan, which option should she choose and why?<br/>$(3)$ When $8 \lt x \lt 40$, which option is more suitable for Xiao Hong and why?

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