
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 \( f(x)=x^{2}-(k^{2}-5ak+3)x+7 \) where \( a, k \in \mathbf{R} \), it is known that for any \( k \in [0,2] \), \( x_{1} \in [k, k+a] \) and \( x_{2} \in [k+2a, k+4a] \), then \( f(x_{1}) \geq f(x_{2}) \). Find the maximum value of the positive real number \( a \).

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