
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 a prime number. Given that the integers \( x_{1}, x_{2}, \cdots, x_{k} \) are pairwise distinct modulo \( p \), and the integers \( y_{1}, y_{2}, \cdots, y_{l} \) are pairwise distinct modulo \( p \), then the sums 
\[ x_{i} + y_{j} \quad (1 \leqslant i \leqslant k, 1 \leqslant j \leqslant l) \]
will have at least \( \min \{p, k+l-1\} \) distinct remainders modulo \( p \).

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