
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 bin has 8 green balls and \( k \) purple balls, where \( k \) is an unknown positive integer. If a ball is drawn at random from the bin, the player wins 5 dollars for a green ball and loses 3 dollars for a purple ball. If the expected amount won from playing the game is $1, find \( k \).

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