
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 all positive real numbers \( a \) such that for any positive real numbers \( t_1, t_2, t_3, t_4 \) satisfying \( t_1 \cdot t_2 \cdot t_3 \cdot t_4 = a^4 \), the following inequality holds:
\[
\frac{1}{\sqrt{1+t_{1}}}+\frac{1}{\sqrt{1+t_{2}}}+\frac{1}{\sqrt{1+t_{3}}}+\frac{1}{\sqrt{1+t_{4}}} \leq \frac{4}{\sqrt{1+a}}.
\]

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