
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  $A$  be the set of the  $16$  first positive integers. Find the least positive integer  $k$  satisfying the condition: In every  $k$ -subset of  $A$ , there exist two distinct  $a, b \in A$  such that  $a^2 + b^2$  is prime.

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