
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 $f(x)=(x-a)^{2}+(2\ln x-2a)^{2}$, where $x > 0, a \in \mathbb{R}$, find the value of the real number $a$ such that there exists $x_{0}$ such that $f(x_{0}) \leqslant \frac{4}{5}$.

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