
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 acute-angled triangle \(ABC\), point \(Q\) is chosen on side \(AC\) such that \(AQ : QC = 1 : 2\). From point \(Q\), perpendiculars \(QM\) and \(QK\) are dropped onto sides \(AB\) and \(BC\) respectively. Additionally, \(BM : MA = 4 : 1\) and \(BK = KC\). Find the ratio \(MK : AC\).

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