
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
Given that in an infinite number of Bernoulli trials, the number of trials $Z$ until both event $A$ and $\overline{A}$ occur follows the distribution $P(Z=k)=P{{(1-P)}^{k-1}}+(1-P){{P}^{k-1}}$, for $k=2,3,\cdots$, calculate the expected value $E(Z)$.

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