
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
Miki's father is saving money in a piggy bank for the family's vacation, adding to it once a week. Miki counts and notes how much money has accumulated every week and looks for patterns in the growth. Let $P_{n}$ denote the amount in the piggy bank in the $n$-th week (in forints). Here are a few observations:

(1) $P_{5} = 2P_{3}$,
(2) $P_{8} = P_{3} + 100$,
(3) $P_{9} = P_{4} + P_{7}$.

"The amount of forints has always been even, but it has never been divisible by 3."

"The number of forints today is a perfect square, and I also noticed that dad increases the deposit each week by the same amount that the third deposit exceeded the second deposit; thus the contents of our piggy bank will never be a perfect square again."

Which week does Miki's last observation refer to, and is Miki's prediction correct?

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