
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
There is a game of picking colored eggs at a market. In an opaque box, there are 9 eggs of the same size and shape, with 3 yellow, 3 red, and 3 blue eggs. The rules of the game are as follows: players first pay a game fee of $10, then randomly pick 3 eggs without replacement. Depending on the colors of the eggs picked, the player may receive a reward. If all 3 eggs picked are of the same color, the player wins $100; if the 3 eggs are all of different colors, the player wins $10; otherwise, there is no reward.  
$(1)$ Let event $A$ be the event that a player picks a yellow egg on the first draw, and event $B$ be the event that the player wins $100 in this game. Find $P(A)$, $P(B)$, and determine if events $A$ and $B$ are independent.  
$(2)$ Decide whether one should play this game and explain the reason.

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