
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  $\triangle ABC$  be an acute triangle,  $H$  is its orthocenter.  $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$  intersect  $\triangle ABC$ 's circumcircle at  $A',B',C'$  respectively. Find the range (minimum value and the maximum upper bound) of  $$ \dfrac{AH}{AA'}+\dfrac{BH}{BB'}+\dfrac{CH}{CC'} $$  

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