
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 game of drawing balls involves a non-transparent paper box containing $6$ identical-sized, differently colored glass balls. Participants pay $1$ unit of fee to play the game once, drawing balls with replacement three times. Participants must specify a color from the box before drawing. If the specified color does not appear, the game fee is forfeited. If the specified color appears once, twice, or three times, the participant receives a reward of $0$, $1$, or $k$ times the game fee ($k \in \mathbb{N}^{*}$), respectively, and the game fee is refunded. Let $X$ denote the participant's profit per game in units of the fee.

(1) Calculate the value of the probability $P(X=0)$;
(2) Determine the minimum value of $k$ such that the expected value of the profit $X$ is not less than $0$ units of the fee.

(Note: Probability theory originated from gambling. Please consciously avoid participating in improper games!)

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