
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
10. Prair picks a three-digit palindrome  $n$  at random. If the probability that  $2n$  is also a palindrome can be expressed as  $\frac{p}{q}$  in simplest terms, find  $p + q$ . (A palindrome is a number that reads the same forwards as backwards; for example,  $161$  and  $2992$  are palindromes, but  $342$  is not.)

11. If two distinct integers are picked randomly between  $1$  and  $50$  inclusive, the probability that their sum is divisible by  $7$  can be expressed as  $\frac{p}{q}$ , where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .

12. Ali is playing a game involving rolling standard, fair six-sided dice. She calls two consecutive die rolls such that the first is less than the second a "rocket." If, however, she ever rolls two consecutive die rolls such that the second is less than the first, the game stops. If the probability that Ali gets five rockets is  $\frac{p}{q}$ , where  $p$  and  $q$  are relatively prime positive integers, find  $p+q$ .

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