
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 person is participating in a prize-winning game activity with the following rules: there is a regular tetrahedron (a triangular pyramid where each face is an equilateral triangle) with the numbers $1$, $2$, $3$, $4$ engraved on its four faces. The tetrahedron is thrown $5$ times, and the number on the face that touches the ground is recorded each time.  
$(1)$ Find the probability that the product of the $5$ numbers on the touching faces is divisible by $4$.  
$(2)$ If a reward of $200$ yuan is given each time the number $3$ is thrown, otherwise a penalty of $100$ yuan is imposed,  
$($i$)$ If person A brings $1000$ yuan to participate in the game, and let $X$ be the amount of money A has after the game, find $E(X)$.  
$($ii$)$ If during the game, A decides to end the game immediately once the money won is not less than $300$ yuan, find the probability that A will end the game in no more than three rounds.

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