
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
The base side $ABCD$ of a regular quadrilateral pyramid $ABCDP$ is equal to $a$, and the lateral edges are equal to $2a$. Consider segments with endpoints on the edges $AD$ and $PC$ that are parallel to the plane $PAB$. 

a) One of these segments is drawn through a point $M$ on edge $AD$ such that $AM:AD=3:4$. Find its length. 

b) Find the minimum length of the considered segments.

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