
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  $S = \{1,..., 999\}$ . Determine the smallest integer  $m$ . for which there exist  $m$  two-sided cards  $C_1$ ,...,  $C_m$  with the following properties: $\bullet$  Every card  $C_i$  has an integer from  $S$  on one side and another integer from  $S$  on the other side. $\bullet$  For all  $x,y \in S$  with  $x\ne y$ , it is possible to select a card  $C_i$  that shows  $x$  on one of its sides and another card  $C_j$  (with  $i \ne j$ ) that shows  $y$  on one of its sides.

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