
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]


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The side of the base \(ABCD\) of a regular pyramid \(SABCD\) is equal to \(a\), and the lateral edge is equal to \(2a\). Consider segments with endpoints on the diagonal \(BD\) of the base and the lateral edge \(SC\), parallel to the plane \(SAD\).

1. One of these segments is drawn through the point \(M\) on the diagonal \(BD\), for which \(DM: DB = 1:3\). Find its length.
2. Find the minimum length of all considered segments.

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