
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 the Bank of Shower, a bored customer lays  $n$  coins in a row. Then, each second, the customer performs ``The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up.

Define the function  $f$  as follows: If there exists some initial arrangement of the coins so that the customer never stops, then  $f(n) = 0$ . Otherwise,  $f(n)$  is the average number of seconds until the customer stops over all initial  configurations. It is given that whenever  $n = 2^k-1$  for some positive integer  $k$ ,  $f(n) > 0$ .

Let  $N$  be the smallest positive integer so that \[
M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right)
\]is a positive integer. If  $M = \overline{b_kb_{k-1}\cdots b_0}$  in base two, compute  $N + b_0 + b_1 + \cdots + b_k$ . 

*Proposed by Edward Wan and Brandon Wang*

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