
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 circle $O$: $x^{2}+y^{2}=1$ intersects with the negative semi-axis of $x$ at point $A$, and $P$ is a point on line $3x+4y-a=0$. A tangent line is drawn from $P$ to circle $O$, intersecting the circle at point $T$. If $PA=2PT$, find the maximum value of $a$.

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