
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
A circle passes through the vertices $A$ and $K$ of triangle $A K T$ and intersects its sides $A T$ and $K T$ at points $C$ and $N$ respectively, with the ratios $A C: C T = 4: 1$ and $T N: N K = 1: 2$. Find $A K$ if $C N = \sqrt{10}$.


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