
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 community in Nanshan District is further implementing the national fitness policy and needs to purchase several pairs of badminton rackets and table tennis paddles for community sports competitions. It is known that purchasing $2$ pairs of badminton rackets and $1$ pair of table tennis paddles costs $330$ yuan; purchasing $5$ pairs of badminton rackets and $2$ pairs of table tennis paddles costs $780$ yuan.  
$(1)$ What is the price of each pair of badminton rackets and table tennis paddles?  
$(2)$ Based on the actual situation of the community, the community plans to use $810$ yuan to purchase several pairs of badminton rackets and table tennis paddles. If exactly $810$ yuan is spent and both types of rackets need to be purchased, what are the possible purchasing plans for the community?

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