
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  $p$  be an odd prime number less than  $10^5$ . Granite and Pomegranate play a game. First, Granite picks a integer  $c \in \{2,3,\dots,p-1\}$ .
Pomegranate then picks two integers  $d$  and  $x$ , defines  $f(t) = ct + d$ , and writes  $x$  on a sheet of paper.
Next, Granite writes  $f(x)$  on the paper, Pomegranate writes  $f(f(x))$ , Granite writes  $f(f(f(x)))$ , and so on, with the players taking turns writing.
The game ends when two numbers appear on the paper whose difference is a multiple of  $p$ , and the player who wrote the most recent number wins. Find the sum of all  $p$  for which Pomegranate has a winning strategy.

*Proposed by Yang Liu*

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