
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
In the acute triangle  $ABC$ , the sum of the distances from the vertices  $B$  and  $C$  to of the orthocenter  $H$  is equal to  $4r,$  where  $r$  is the radius of the circle inscribed in this triangle. Find the perimeter of triangle  $ABC$  if it is known that  $BC=a$ .

(Gryhoriy Filippovskyi)

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