
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
In a lottery game, the host randomly selects one of the four identical-looking empty boxes numbered $1$, $2$, $3$, $4$, puts a prize inside, and then closes all four boxes. The host knows which box the prize is in. The rules of the game are that the host asks the player to choose one of the four boxes, and if the prize is in that box, the player wins the prize. Player A chooses box $2$. Before opening box $2$, the host first opens one of the other three empty boxes. According to the rules of the game, the host will randomly open an empty box other than the one chosen by player A.  
$(1)$ Calculate the probability that the host opens box $4$.  
$(2)$ After the host opens box $4$, player A is given a chance to choose again. Should he stick with box $2$ or switch to box $1$ or box $3$ to maximize his chances of winning the prize?

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