
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
\[ f(x) = x^2 - (k^2 - 5ak + 3)x + 7 \quad (a, k \in \mathbb{R}) \]
for any \( k \in [0, 2] \), if \( x_1, x_2 \) satisfy
\[ x_1 \in [k, k+a], \quad x_2 \in [k+2a, k+4a], \]
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