
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
<u>Set 3</u>**3.1** Annie has  $24$  letter tiles in a bag;  $8$  C’s,  $8$  M’s, and  $8$  W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?**3.2** Let  $T$  be the answer from the previous problem. Charlotte is initially standing at  $(0, 0)$  in the coordinate plane. She takes  $T$  steps, each of which moves her by  $1$  unit in either the  $+x$ ,  $-x$ ,  $+y$ , or  $-y$  direction (e.g. her first step takes her to  $(1, 0)$ ,  $(1, 0)$ ,  $(0, 1)$  or  $(0, -1)$ ). After the T steps, how many possibilities are there for Charlotte’s location?**3.3** Let  $T$  be the answer from the previous problem, and let  $S$  be the sum of the digits of  $T$ . Francesca has an unfair coin with an unknown probability  $p$  of landing heads on a given flip. If she flips the coin  $S$  times, the probability she gets exactly one head is equal to the probability she gets exactly two heads. Compute the probability  $p$ .


PS. You should use hide for answers. 

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