
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  $n$  be a positive integer. A *Japanese triangle* consists of  $1 + 2 + \dots + n$  circles arranged in an equilateral triangular shape such that for each  $i = 1$ ,  $2$ ,  $\dots$ ,  $n$ , the  $i^{th}$  row contains exactly  $i$  circles, exactly one of which is coloured red. A *ninja path* in a Japanese triangle is a sequence of  $n$  circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with  $n = 6$ , along with a ninja path in that triangle containing two red circles.
[asy]
unitsize(7mm);
path q=(3,-3sqrt(3))--(-3,-3sqrt(3));
filldraw(shift(0*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
filldraw(shift(1*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(1*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(2*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(2*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(2*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(3*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(3*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(3*dir(240)+2*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(3*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+0*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(4*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(4*dir(240)+3*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(4*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
filldraw(shift(5*dir(240)+0*dir(0))*scale(0.5)*unitcircle,lightred);
draw(shift(5*dir(240)+1*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+2*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+3*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+4*dir(0))*scale(0.5)*unitcircle);
draw(shift(5*dir(240)+5*dir(0))*scale(0.5)*unitcircle);
draw((0,0)--(1/2,-sqrt(3)/2)--(0,-sqrt(3))--(1/2,-3sqrt(3)/2)--(0,-2sqrt(3))--(-1/2, -5sqrt(3)/2),linewidth(1.5));
draw(q,Arrows(TeXHead, 1));
label(" $n = 6$ ", q, S);
[/asy]
In terms of  $n$ , find the greatest  $k$  such that in each Japanese triangle there is a ninja path containing at least  $k$  red circles.

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