
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
There are two piles of candies, and two players take turns performing the following operation: In each turn, a player can take one entire pile. If the remaining pile has more than one candy, it can be arbitrarily divided into two piles; if the remaining pile has only one candy, the player takes that candy too and wins the game. Suppose initially, the two piles have $p$ and $q$ candies respectively. What necessary and sufficient condition should $(p, q)$ satisfy for the first player to guarantee a win?

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