
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 empty boxes numbered $1$, $2$, $3$, $4$, puts a prize inside, and then closes all four boxes. The host knows which box contains the prize. When a participant chooses a box, before opening the chosen box, the host randomly opens another box without the prize and asks the participant if they would like to change their selection to increase the chances of winning. Let $A_{i}$ represent the event that box $i$ contains the prize $(i=1,2,3,4)$, and let $B_{i}$ represent the event that the host opens box $i$ $(i=2,3,4)$. Now, if it is known that the participant chose box $1$, then $P(B_{3}|A_{2})=$______; $P(B_{3})=______.$

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