
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
In a checked  $ 17\times 17$  table,  $ n$  squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least  $ 6$  of the squares in some line are black, then one can paint all the squares of this line in black. 
Find the minimal value of  $ n$  such that for some initial arrangement of  $ n$  black squares one can paint all squares of the table in black in some steps.

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