
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
Given that the sequence $\{a\_n\}$ is a positive arithmetic sequence satisfying $\frac{1}{a\_1} + \frac{4}{a\_{2k-1}} \leqslant 1$ (where $k \in \mathbb{N}^*$, and $k \geqslant 2$), find the minimum value of $a\_k$.

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