
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
In Prime Land, there are seven major cities, labelled  $C_0$ ,  $C_1$ , \dots,  $C_6$ . For convenience, we let  $C_{n+7} = C_n$  for each  $n=0,1,\dots,6$ ; i.e. we take the indices modulo  $7$ .  Al initially starts at city  $C_0$ . 

Each minute for ten minutes, Al flips a fair coin.  If the coin land heads, and he is at city  $C_k$ , he moves to city  $C_{2k}$ ; otherwise he moves to city  $C_{2k+1}$ .  If the probability that Al is back at city  $C_0$  after  $10$  moves is  $\tfrac{m}{1024}$ , find  $m$ .

*Proposed by Ray Li*

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