
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 kite interest group at a certain middle school decided to hold a blind box kite competition. The competition adopts a scoring system to select the winner. The available kites are hard-wing kites, soft-wing kites, series kites, flat kites, and three-dimensional kites, totaling $5$ types of kites. The kites are placed in blind boxes for participants to draw. Each participant who draws a hard-wing kite or a soft-wing kite will score $1$ point and fly the kite. Drawing a series kite, a flat kite, or a three-dimensional kite will score $2$ points and fly the kite. The results of drawing kites are independent of each other, and only one kite can be drawn each time. The probability for each participant to draw a hard-wing kite or a soft-wing kite is $\frac{2}{5}$, and the probability to draw the other $3$ types of kites is $\frac{3}{5}$.

$(1)$ If participant A draws blind boxes consecutively $2$ times, the total score is $X$ points. Find the distribution and expectation of $X$.

$(2)$ Assuming participant B can continuously draw blind boxes, meaning the number of times to draw blind boxes can be any number from $1, 2, 3, \ldots$. Let the probability of accumulating a total score of $n$ points by B be $P(n)$. When $n \geq 3$, find $P(n)$.

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