
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
 How many (possibly empty) sets of lattice points  $\{P_1, P_2, ... , P_M\}$ , where each point  $P_i =(x_i, y_i)$  for  $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line  $P_iP_j$  is positive for each  $1 \le i < j \le M$  ? An infinite slope, e.g.  $P_i$  is vertically above  $P_j$  , does not count as positive.

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