
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
Let  $T=\text{TNFTPP}$ . Fermi and Feynman play the game  $\textit{Probabicloneme}$  in which Fermi wins with probability  $a/b$ , where  $a$  and  $b$  are relatively prime positive integers such that  $a/b<1/2$ . The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play  $\textit{Probabicloneme}$  so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is  $(T-332)/(2T-601)$ .  Find the value of  $a$ .

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