
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
Given is a acute angled triangle  $ABC$ . The lengths of the altitudes from  $A, B$  and  $C$  are successively  $h_A, h_B$  and  $h_C$ . Inside the triangle is a point  $P$ . The distance from  $P$  to  $BC$  is  $1/3 h_A$  and the distance from  $P$  to  $AC$  is  $1/4 h_B$ . Express the distance from  $P$  to  $AB$  in terms of  $h_C$ .

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