
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 acute triangle has two of its altitudes dividing the sides into segments of lengths $6, 4, x$ and $3$ units, respectively. Determine the value of $x$. [asy]
defaultpen(linewidth(0.7)); size(75);
pair A = (0,0);
pair B = (1,0);
pair C = (52/136,119/136);
pair D = foot(B, A, C);
pair E = /*foot(A,B,C)*/ (72*B+(119-72)*C)/(119);
draw(A--B--C--cycle);
draw(B--D);
draw(A--E);
draw(rightanglemark(A,D,B,2));
draw(rightanglemark(A,E,B,2));
label("$4$",(C+D)/2,WNW+(0,0.5));
label("$6$",(A+D)/2,NW);
label("$3$",(C+E)/2,E);
label("$x$",(B+E)/2,NE);
[/asy]

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