
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
Nowadays, "online group buying" has become a very common way of shopping in people's lives. A merchant is launching an online New Year's goods flash sale activity before the Spring Festival, allowing the community to enjoy convenience. On February 8th, the merchant conducted an online flash sale for each free-range ecological chicken, with the flash sale price of each chicken being $15 less than before the activity. Wang Dama bought twice as many ecological chickens for $120 during the event as she did for $90 before the event.  
$(1)$ Find the original selling price of each free-range ecological chicken.  
$(2)$ Due to the success of the flash sale on February 8th, the merchant decided to hold a limited-time flash sale at 12:00 on the last day before the New Year. The flash sale price of each free-range ecological chicken remains the same, and the flash sale lasts for 1 hour, with a limit of 1 chicken per person. At 12:00 on the day of the event, 50 people made online purchases. Subsequently, the number of new buyers increased by $m\%$ every 30 minutes. By 13:00, the total sales of the flash sale activity amounted to $5460. Find the value of $m$.

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