
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 merchant made some cards to promote a certain product. The cards come in $3$ different colors, and each time a customer makes a purchase meeting the full amount, they will randomly receive $1$ card of a certain color. Collecting $3$ cards of the same color allows the customer to exchange for one unit of the product.<br/>$(1)$ Find the probability that a customer, after making full purchases $4$ times, still has not collected $3$ cards of the same color;<br/>$(2)$ Let $X$ be the number of times a customer makes full purchases before collecting exactly $3$ cards of the same color. Find the distribution and expectation of $X$.

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