
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
Let the real numbers \( a \) and \( b \) satisfy \( a = x_1 + x_2 + x_3 = x_1 x_2 x_3 \) and \( ab = x_1 x_2 + x_2 x_3 + x_3 x_1 \), where \( x_1, x_2, x_3 > 0 \). Then the maximum value of \( P = \frac{a^2 + 6b + 1}{a^2 + a} \) is ______.

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