
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
Triangles $ABC$ and $AGH$ are such that $B$ is the midpoint of $\overline{GH}.$ Also, $AB = GH = 1,$ $BC = 10,$ $CA = \sqrt{97},$ and
\[\overrightarrow{AB} \cdot \overrightarrow{AG} + \overrightarrow{AC} \cdot \overrightarrow{AH} = 6.\]
Find the cosine of the angle between vectors $\overrightarrow{GH}$ and $\overrightarrow{BC}.$

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