
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 circle \(\Gamma: x^{2} + y^{2} = 1\) with two intersection points with the \(x\)-axis as \(A\) and \(B\) (from left to right), \(P\) is a moving point on circle \(\Gamma\). Line \(l\) passes through point \(P\) and is tangent to circle \(\Gamma\). A line perpendicular to \(l\) passes through point \(A\) and intersects line \(BP\) at point \(M\). Find the maximum distance from point \(M\) to the line \(x + 2y - 9 = 0\).

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