
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 real numbers \( a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5 \leq a_6 \) satisfying \( a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 10 \) and \( (a_1 - 1)^2 + (a_2 - 1)^2 + (a_3 - 1)^2 + (a_4 - 1)^2 + (a_5 - 1)^2 + (a_6 - 1)^2 = 6 \), what is the largest possible value of \( a_6 \)?

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