
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 a triangle with a right angle  $\angle ABC$ .  Let  $D$  be the midpoint of  $\overline{BC}$ , let  $E$  be the midpoint of  $\overline{AC}$ , and let  $F$  be the midpoint of  $\overline{AB}$ .  Let  $G$  be the midpoint of  $\overline{EC}$ .  One of the angles of  $\triangle DFG$  is a right angle.  What is the least possible value of  $\frac{BC}{AG}$ ?

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