
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
The altitudes in triangle  $ABC$  have lengths 10, 12, and 15. The area of  $ABC$  can be expressed as  $\tfrac{m\sqrt n}p$ , where  $m$  and  $p$  are relatively prime positive integers and  $n$  is a positive integer not divisible by the square of any prime. Find  $m + n + p$ . 

[asy]
import olympiad;
size(200);
defaultpen(linewidth(0.7)+fontsize(11pt));
pair A = (9,8.5), B = origin, C = (15,0);
draw(A--B--C--cycle);
pair D = foot(A,B,C), E = foot(B,C,A), F = foot(C,A,B);
draw(A--D^^B--E^^C--F);
label(" $A$ ",A,N);
label(" $B$ ",B,SW);
label(" $C$ ",C,SE);
[/asy]

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