
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
Let $0 < \epsilon \leq 1 / 4$, and let $\mathcal{P}=\{C_{0}, C_{1}, \ldots, C_{k}\}$ be a partition of $V$, where the order of the exceptional set $C_{0}$ satisfies $\left|C_{0}\right| \leq \epsilon n$ and $\left|C_{1}\right|=\cdots=\left|C_{k}\right|=: c$. If $\mathcal{P}$ is not $\epsilon$-regular, then there exists a partition $\mathcal{P}^{\prime}=\{C_{0}^{\prime}, C_{1}^{\prime}, \ldots, C_{\ell}^{\prime}\}$ of $V$, where the exceptional set $C_{0}^{\prime}$ satisfies $\left|C_{0}^{\prime}\right| \leq \left|C_{0}\right| + n / 2^{k}$ and $k \leq \ell \leq k 4^{k}$, and all other sets $C_{i}^{\prime}$ have the same number of elements, with 
$$
q(\mathcal{P}^{\prime}) \geq q(\mathcal{P}) + \epsilon^{5} / 2.
$$

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