
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  $ABC$  have side lengths  $3$ ,  $4$ , and  $5$ . Let  $P$  be a point inside  $ABC$ . What is the minimum sum of lengths of the altitudes from  $P$  to the side lengths of  $ABC$ ?

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