
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  $T = (a,b,c)$  be a triangle with sides  $a,b$  and  $c$  and area  $\triangle$ .  Denote by  $T' = (a',b',c')$  the triangle whose sides are the altitudes of  $T$  (i.e.,  $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by  $\triangle '$ .  Similarly, let  $T'' = (a'',b'',c'')$  be the triangle formed from the altitudes of  $T'$ , and denote its area by  $\triangle ''$ . Given that  $\triangle ' = 30$  and  $\triangle '' = 20$ , find  $\triangle$ .

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