
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
From an urn containing 3 blue balls and 2 red balls, balls are drawn sequentially without replacement according to a random selection scheme. Find the probability $P_{k}$ that a red ball appears for the first time at the $k$-th trial ($k=1,2,3,4$).

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