
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
Let $P_n$ denote the probability that Tom, Dick, and Harry will have flipped their coins $n$ times with all showing the same number of heads on each flip. Given that the probability of getting a head is $\frac{1}{3}$ and the probability of getting a tail is $\frac{2}{3}$, calculate $P_1+P_2+P_3+P_4+P_5$.

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