
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 side of the base $ABCD$ of a regular prism $ABCD A_1B_1C_1D_1$ is $2a$, and the lateral edge is $a$. We consider segments with endpoints on the diagonal $AD_1$ of the face $AA_1D_1D$ and the diagonal $DB_1$ of the prism, parallel to the plane $AA_1B_1B$.

a) One of these segments passes through a point $M$ on the diagonal $AD_1$ such that $AM : AD_1 = 2 : 3$. Find its length.

b) Find the minimum length of all the considered segments.

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