
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
Triangle  $ ABC$  has fixed vertices  $ B$  and  $ C$ , so that  $ BC \equal{} 2$  and  $ A$  is variable. Denote by  $ H$  and  $ G$  the orthocenter and the centroid, respectively, of triangle  $ ABC$ . Let  $ F\in(HG)$  so that  $ \frac {HF}{FG} \equal{} 3$ . Find the locus of the point  $ A$  so that  $ F\in BC$ .

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