
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
Find the smallest positive number \( a \) such that there exists a positive number \( b \) where for all \( x \in [0, 1] \), the inequality \(\sqrt{1-x} + \sqrt{1+x} \leq 2 - b x^a\) holds. For the obtained value of \( a \), determine the largest positive number \( b \) that satisfies the above inequality.

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