
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
The sum of four positive numbers is 4, and the sum of their squares is 8. Determine the maximum value of the largest number among them. Reformulating this, let $a \geqslant b \geqslant c \geqslant d > 0,$ and satisfy

$$
a + b + c + d = 4, \quad a^{2} + b^{2} + c^{2} + d^{2} = 8,
$$

find $\max a = ?$

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