
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 the point $M(m, m^2)$ and $N(n, n^2)$, where $m$ and $n$ are the two distinct real roots of the equation $\sin\theta \cdot x^2 + \cos\theta \cdot x - 1 = 0 (\theta \in R)$. If the maximum distance from a point on the circle $O: x^2 + y^2 = 1$ to the line $MN$ is $d$, and the positive real numbers $a$, $b$, and $c$ satisfy the equation $abc + b^2 + c^2 = 4d$, determine the maximum value of $\log_4 a + \log_2 b + \log_2 c$.

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