
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 $x$ be a positive real number and $\theta$ a real number such that $0 \leq \theta \leq \pi$. Find the maximum possible value of $$\frac{x^2 + 2 - \sqrt{x^4 + 4\sin^2\theta}}{x}.$$

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