
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
An L-shape is one of the following four pieces, each consisting of three unit squares:

[asy]
size(300);
defaultpen(linewidth(0.8));
path P=(1,2)--(0,2)--origin--(1,0)--(1,2)--(2,2)--(2,1)--(0,1);
draw(P);
draw(shift((2.7,0))*rotate(90,(1,1))*P);
draw(shift((5.4,0))*rotate(180,(1,1))*P);
draw(shift((8.1,0))*rotate(270,(1,1))*P);
[/asy]
A  $5\times 5$  board, consisting of  $25$  unit squares, a positive integer  $k\leq 25$  and an unlimited supply of L-shapes are given. Two players A and B, play the following game: starting with A they play alternatively mark a previously unmarked unit square until they marked a total of  $k$  unit squares.

We say that a placement of L-shapes on unmarked unit squares is called  $\textit{good}$  if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board.
 
B wins if every  $\textit{good}$  placement of L-shapes leaves uncovered at least three unmarked unit squares.  Determine the minimum value of  $k$  for which B has a winning strategy.

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