
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 function $f(x)=2a\ln x-x^{2}+1$:
1. If $a=1$, find the interval where the function $f(x)$ is monotonically decreasing.
2. If $a \gt 0$, find the maximum value of the function $f(x)$ on the interval $\left[1,+\infty \right)$.
3. If $f(x)\leqslant 0$ holds true on the interval $\left[1,+\infty \right)$, find the maximum value of $a$.

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