
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  $N$  be the number of sequences of words (not necessarily grammatically correct) that have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMATHBONANZA. Here is an example of a possible sequence:
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        N, NA, NZA, BNZA, BNAZA, BONAZA, BONANZA, CBONANZA, CABONANZA, CAMBONANZA, CAMABONANZA, CAMAHBONANZA, CCAMAHBONANZA, CCAMATHBONANZA.
[/center]
    Estimate  $\frac{N}{12!}$ . An estimate of  $E>0$  earns  $\max(0,4-2\max(A/E,E/A))$  points, where  $A$  is the actual answer. An estimate of  $E=0$  earns  $0$  points.


*2021 CCA Math Bonanza Lightning Round #5.3*

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