
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
Define $b-a$ as the "length" of the set $\{x \mid a \leqslant x \leqslant b\}$. Let $M=\left\{x \mid m \leqslant x \leqslant m+\frac{3}{4}\right\}$ and $N=\left\{x \mid n-\frac{1}{3} \leqslant x \leqslant n\right\}$, with both $M$ and $N$ being subsets of $\{x \mid 0 \leqslant x \leqslant 1\}$. Find the minimum "length" of the set $M \cap N$.

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