
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 \( a, b, c \in \mathbb{R}^{+} \) and \( a b c + a + c = b \). Determine the maximum value of \( P = \frac{2}{a^{2} + 1} - \frac{2}{b^{2} + 1} + \frac{3}{c^{2} + 1} \).

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