
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
p1. For what values of  $a$  is  $(5 a + 1)(3a + 2)$  divisible by  $15$ ?


p2. Of the four-digit positive integers that are multiple of  $9$ , how many are there that have all their digits different from zero and different from each other?


p3. A square with side  $5$  is divided into  $25$  unit squares by lines parallel to the sides. Let A be the set of the  $16$  interior points, which are vertices of the unit squares, but which are not on the sides of the initial square. What is the largest number of points in  $A$  that can be chosen so that any three of them not vertices of a right isosceles triangle?


p4. In a parallelogram  $ABCD$ ,  $BC = 2AB$ . The bisector of angle  $B$  intersects the extension of segment  $CD$  at  $Q$  and the bisector of angle  $A$  intersects segment  $BD$  at  $M$  and segment  $BC$  at  $P$ . If  $PQ = 6$ ,  find the length of the segment  $MB$ .


p5. Find the number of ways to write nonnegative integers in each box on a board of dimension  $ n \times n$ ,  so that the sum of the numbers in each row and each column is equal to  $3$  and in each row and in each column there are only one or two numbers different from zero.

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