
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
For a fixed \(\theta \in \left(0, \frac{\pi}{2}\right)\), find the smallest positive number \(a\) that satisfies the following two conditions:
(i) \(\frac{\sqrt{a}}{\cos \theta} + \frac{\sqrt{a}}{\sin \theta} > 1\);
(ii) There exists \(x \in \left[1 - \frac{\sqrt{a}}{\sin \theta}, \frac{\sqrt{a}}{\cos \theta}\right]\) such that
\[
\left[(1-x) \sin \theta - \sqrt{a - x^2 \cos^2 \theta}\right]^2 + \left[x \cos \theta - \sqrt{a - (1-x)^2 \sin^2 \theta}\right]^2 \leqslant a.
\]

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