
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
**p10.** Three rectangles of dimension  $X \times 2$  and four rectangles of dimension  $Y  \times 1$  are the pieces that form a rectangle of area  $3XY$  where  $X$  and  $Y$  are positive, integer values. What is the sum of all possible values of  $X$ ?**p11.** Suppose we have a polynomial  $p(x) = x^2 + ax + b$  with real coefficients  $a + b = 1000$  and  $b > 0$ . Find the smallest possible value of  $b$  such that  $p(x)$  has two integer roots.**p12.** Ten square slips of paper of the same size, numbered  $0, 1, 2, ..., 9$ , are placed into a bag. Four of these squares are then randomly chosen and placed into a two-by-two grid of squares. What is the probability that the numbers in every pair of blocks sharing a side have an absolute difference no greater than two?


PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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