
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
(1) Find the equation of the line that passes through the point $(2,3)$ and has equal intercepts on the $x$-axis and $y$-axis.

(2) Given a circle $(x^2)-4x-4+(y^2)=0$ and a point $P(x,y)$ on the circle, find the maximum value of $x^2+y^2$.

(3) Given a circle $(x^2)+(y^2)=4$ and a point outside the circle $P(-2,-3)$, find the equation of the tangent line to the circle that passes through point $P$.

(4) Let $l$ be a line, and $\alpha$, $\beta$, $\gamma$ be three distinct planes. Consider the following four propositions: 
   - (i) If $\alpha \perp \gamma$ and $\beta \perp \gamma$, then $\alpha \perp \beta$.
   - (ii) If $\alpha \perp \gamma$ and $\beta \parallel \gamma$, then $\alpha \perp \beta$.
   - (iii) If $l \parallel \alpha$ and $l \perp \beta$, then $\alpha \perp \beta$.
   - (iv) If $l \parallel \alpha$, then $l$ is parallel to all lines within $\alpha$.
   
   Indicate the correct propositions by their numbers.

(5) Through point $P(4,9)$, draw two tangents $PA$ and $PB$ to the circle $(x^2)+(y^2)=4$ (with $A$ and $B$ being the points of tangency). Find the equation of line $AB$.

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