
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 parallelogram \( ABCD \), which is not a rhombus, the ratio of the lengths of the diagonals \( AC \) and \( BD \) is given as \( k \). Let ray \( AM \) be symmetric to ray \( AD \) with respect to line \( AC \), and let ray \( BM \) be symmetric to ray \( BC \) with respect to line \( BD \). \( M \) is the common point of rays \( AM \) and \( BM \). Find the ratio \( AM : BM \).

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