
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  $\triangle A_1B_1C_1$  be an equilateral triangle of area  $60$ .  Chloe constructs a new triangle  $\triangle A_2B_2C_2$  as follows.  First, she flips a coin.  If it comes up heads, she constructs point  $A_2$  such that  $B_1$  is the midpoint of  $\overline{A_2C_1}$ .  If it comes up tails, she instead constructs  $A_2$  such that  $C_1$  is the midpoint of  $\overline{A_2B_1}$ .  She performs analogous operations on  $B_2$  and  $C_2$ .  What is the expected value of the area of  $\triangle A_2B_2C_2$ ?

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