
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
Points \( M \) and \( N \) are located on side \( BC \) of triangle \( ABC \), and point \( K \) is on side \( AC \), with \( BM : MN : NC = 1 : 1 : 2 \) and \( CK : AK = 1 : 4 \). Given that the area of triangle \( ABC \) is 1, find the area of quadrilateral \( AMNK \).

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