
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 an intelligence test, there are two independent questions A and B. The scoring rules are as follows: if the test taker answers question A correctly, they will receive a score of $a$; if they answer question B correctly, they will receive a score of $b$. No points are awarded for incorrect answers. The test taker can choose which question to answer first, but they can only proceed to the second question if they answer the first one correctly; otherwise, the test ends. If you are the test taker and assume that your probabilities of correctly answering questions A and B are $P_1$ and $P_2$ respectively:  
(Ⅰ) If $P_1 = \frac{1}{2}$ and $P_2 = \frac{1}{3}$, how should you choose which question to answer first based on the point values of the questions?  
(Ⅱ) Given $a=10$, $b=20$, and $P_1 = \frac{2}{5}$, from a statistical perspective, within what range of $P_2$ should you choose to answer question A first so that the average score of choosing A first is not lower than that of choosing B first?

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