
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  $ABC$  be a triangle with side lengths  $AB=6, AC=7,$  and  $BC=8.$  Let  $H$  be the orthocenter of  $\triangle ABC$  and  $H'$  be the reflection of  $H$  across the midpoint  $M$  of  $BC.$   $\tfrac{[ABH']}{[ACH']}$  can be expressed as  $\frac{p}{q}$ . Find  $p+q$ .

*2022 CCA Math Bonanza Individual Round #14*

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