
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
If $A B C$ is a triangle, $O$ is the center of the circumscribed circle with radius $R$, and $H$ is the orthocenter, show the following:

1. $\overrightarrow{O A} \cdot \overrightarrow{O B}=R^{2}-\frac{c^{2}}{2}$;
2. $\overrightarrow{O H}=\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}$;
3. $O H^{2}=9 R^{2}-a^{2}-b^{2}-c^{2}$.

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